College Panda
The College
SAT
SAT Math
Advanced
and Workbook
Advanced Guide
Guide and
Workbook
2nd
2nd Edition
Edition
Copyright
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Introduction
Introduction
The best
best way
to do
do well
well on
on any
any test
test is
is to
to be
be experienced
experienced with
material. Nowhere
is this
this more
true than
than on
on
The
way to
with the
the material.
Nowhere is
more true
the
SAT, which
which is standardized
standardized to
to repeat
repeat the
the same
same question
question types
and again.
again. The
The purpose
this book
book is
the SAT,
types again
again and
purpose of this
to teach
teach you
you the
the concepts
concepts and
and battle-tested
battle‐tested approaches
approaches you
questions types
types.. If
it's
you need
need to know
know for all these
these questions
If it's
n
o t in
in this
this book,
book, it's
it’s not
not on
on the
the test.
test. The
The goal
goal is
is for every
every SAT question
be a
simple reflex,
you
not
question to be
a simple
reflex, something
something you
know
how to
to handle
handle instinctively
instinctively because
because you've
you’ve seen
seen it so
times before.
before.
know how
so many
many times
won’t find any
any cheap
cheap tricks
tricks in
in this
this book,
book, simply
simply because
because there
there aren’t
that work
work consistently.
consistently. Don't
Don’t buy
You won't
aren't any
any that
buy
into the
the idea
idea that
that you
you can
can improve
improve your
your score
score significantly
significantly without
work..
into
without hard
hard work
Format
Format of the Test
There
math sections
be done
minutes without
without aa
There are
are two
t w o math
sections on
on the
the SAT. The
The first contains
contains 20 questions
questions to be
done in 25
25 minutes
calculator.
and a
a calculator
calculator. The
The second
second contains
contains 38
38 questions
questions to be
be done
done in 55
55 minutes
minutes and
calculator is permitted
permitted..
Some
made sure
accurately divide
the practice
practice questions
Some topics
topics only
only show
show up
up in the
the calculator
calculator section.
section. I've
I’ve made
sure to accurately
divide the
questions
into
into non-calculator
non-calculator and
and calculator
calculator components.
components.
How
How to Read
Read this
this Book
Book
For
beginning to end.
chapter was
was
For aa complete
complete understanding,
understanding, this
this book
book is best
best read
read from
from beginning
end. That
That being
being said,
said, each
each chapter
written
possible. After
all, you
may already
already be
written to be
be independent
independent of the others
others as
as much
much as
aspossible.
After all,
you may
be proficient
proficient in some
some
topics yet weak
others. If so, feel free to
on the
the chapters
chapters that
are most
most relevant
topics
weak in others.
to jump
jump around,
around, focusing
focusing on
that are
relevant to
your
your improvement.
improvement.
All chapters
won't master
master the
the material
material until
think through
through the
chapters come
come with
with exercises.
exercises. Do them
them.. You won’t
until you
you think
the
questions
yourself.
If
you
get
stuck
on
a
question,
give
yourself
a
few
minutes
to
figure
it
out.
If
you’re
que stions yourself.
you
stuck on a question, give yourself a
minutes
figure out. If you're
still stuck,
stuck, then
then look
look at
at the
solution and
and take
take the
the time
time to fully understand
Then circle
circle the
the question
number
still
the solution
understand it. Then
question number
or make
make aa note
note of it somewhere
somewhere so
so that
that you
you can
can redo
redo the
the question
Revisiting questions
missed is
question later.
later. Revisiting
questions you’ve
you've missed
the best
best way
way to improve
improve your
your score.
score.
About
About the
the Author
Nielson Phu
Phu graduated
graduated from New
New York University,
University, where
where he
he studied
science. He
He has
obtained perfect
perfect
Nielson
studied actuarial
actuarial science.
has obtained
scores on the SAT and
and on the
the SAT math
math subject
subject test.
test. As aa teacher,
he has
has helped
helped hundreds
students
scores
teacher, he
hundreds of students
throughout Boston
Boston and
and Hong
Hong Kong
Kong perform
perform better
better on
on standardized
standardized tests.
Although he
he continues
pursue
throughout
tests. Although
continues to pursue
his interests
interests in education,
education, he
n o w an
an engineer
engineer in the
the Boston
Boston area.
his
he is now
area.
THE COLLEGE
A
COLLEGE PAND
PANDA
Table of Contents
Contents
1
Exponents
Exponents & Radicals
Radicals
7
Laws
Laws of exponents
exponents
Evalua
ting expressions
nen ts
Evaluating
expressions with
with expo
exponents
Solving
equations
with
exponents
Solving equations with exponents
Simplifying
Simplifying square
square roots
roots
2
15
15
Percent
Percent
Percent
Percent change
change
Compound
Compound interest
interest
Percent
problems
Percent word
word problems
33
Exponential
Exponential vs
vs.. Linear
Linear Growth
Growth
23
Linear
growth and
and decay
decay
Linear growth
Exponentia
Exponentiall growth
growth and
and decay
decay
Positive
tion
Positive and
and negative
negative associa
association
4
Rates
Rates
32
Conversion factors
factors
Conversion
5
6
6
Ratio
Ratio & Proportion
Proportion
38
38
44
44
Expressions
Expressions
Combining
like terms
terms
Combining like
Expansion
and factoring
Expans
ion and
factoring
Combining, dividing
dividing,, and
and splitting
splitting fractions
fractions
Combining,
7
8
Constructing Models
Models
Constructing
52
Manipulating &
& Solving
Solving Equations
Equations
Manipulating
57
Common mistakes
mistakes to avoid
avoid
Common
isolating variables
variables
Tools for isolating
Strategies for solving
solving complicated
complicated equations
equations
Strategies
9
More Equation
Equation Solving
Solving Strategies
Strategies
More
71
Matching coefficients
coefficients
Matching
lnfinitely many
many solutions
solutions
Infinitely
solutions
No soluti
ons
Clearing denominators
denominators
Clearing
10
10
Systems of
of Equations
Equations
Systems
78
Substitution
Substitution
Elimination
Elimination
Systems with
with no solutions
solutions and
and infinite
infinite solutions
solutions
Systems
problems
Word problems
More comp
complex
systems
More
lex systems
Graphs of systems
systems of equations
equations
Graphs
4
.r
THE COLLEGE
PANDA
COLLEGE PANDA
11
Inequalities
Inequalities
91
inequalities
solve inequalities
H o w to solve
How
problems
word problems
Inequality
Inequality word
inequalities
Graphs
Graphs of inequalities
12
13
13
14
14
Problems
Word Problems
100
Problems
& Maximum
Minimum
M i n i m u m 8:
Maximum Word Problems
109
Lines
Lines
117
y-intercept
and y-intercept
Slope and
Slope
point-slope form
and point-slope
form and
lines: slope-intercept
Equations of lines:
Equations
slope-intercept form
form
intersection of two
the intersection
Finding the
Finding
t w o lines
lines
lines
perpendicular lines
and perpendicular
Parallel and
Parallel
vertica l lines
and vertical
Horizontal and
Horizontal
lines
15
15
16
126
132
132
Linear Models
Interpreting
Models
Interpreting Linear
Functions
Functions
function?
a function?
What is a
What
undefined?
function undefined?
When is a function
When
functions
Composite functions
Composite
Finding
solutions to a
function
a function
the solutions
Finding the
graphs
function graphs
Identifying function
Identifying
transformations
Function transformations
Function
17
146
146
Quadratics
Quadratics
Tactics for finding
roots
finding the roots
Completing
the
square
square
the
Completing
The vertex
form
vertex form
and vertex
ver tex and
The
discriminant
The discriminant
Quadratic
models
Quadratic models
18
18
Synthetic
Division
Synthetic Division
161
Performing
synthetic division
division
Performing synthetic
Equivalent expressions
expressions
Equivalent
The remainder
theorem
remainder theorem
19
20
21
21
Complex
Numbers
Complex Numbers
Absolute
Value
Absolute Value
Angles
Angles
170
170
174
180
180
Exterior
theorem
angle theorem
Exterior angle
Parallel
lines
Parallel lines
Polygons
Polygons
5
THE COLLEGE PANDA
22
Triangles
187
Isosceles and
and equilateral
equi lateral triangles
lsosceles
triangles
Right triangles
Right
triangles
Specia l right
Special
right triangles
triangles
Similar
Similar triangles
triangles
Parallel
Parallel Lines and
Proportionality
and Proportionality
Radians
Radians
23
Circles
Circles
207
Area and
and circumference
circumference
Arc length
length
Area of a sector
sector
Centra l and
and inscribed
inscribed angles
Central
ang les
Equa
tion s of circles
Equations
24
Trigonometry
Trigonometry
216
and tangent
tangent
Sine, cosine,
cosine, and
Trigonometric identities
Trigonometric
identities
Evaluating trigonometric
Evaluating
trigonometric expressions
expressions
25
26
27
Reading Data
Reading
Data
Probability
Probability
225
234
244
Statistics
Statistics II
Mean,
and mode
Mean, median,
median, and
mode
Range
standard
Range and
and standard deviation
deviation
Histograms
Histograms and
and dot
dot plots
plots
problems involving
involving averages
averages
Word problems
Boxplots
Boxplots
28
Statistics II
II
Statistics
254
Statistical
sampling
Statistica l sampling
Using and
and interpreting
interpreting the
the line
line of best fit
Using
Margin
error
Margin of error
Confidence
intervals
Confidence intervals
Experimental
Experimental design
design and
and conclusions
conclusions
29
Volume
Volume
267
30
Answers
Answers to the Exercises
Exercises
272
6
Ex
ponents & Ra
Exponents
Radicals
dicals
Here
are the
Here are
the laws
laws of exponents
exponents you
you should
should know:
know:
Example
Example
Law
Law
1
xx1z
= Xx
F
1
331z
= 33
,
L
\U_
XO
=
11
A
30_
3D= 11
1
m 0n
111In I 1 111
11
xX",
34 35 Z 39
34.
35 = 39
· x" = x +
x111
if:
:
_=
7
27
Z 34
34
37 =
xm‐n
XIII -II
x"
33
4
(32)-l
: 38
38
(32)4 =
(XVII)
Z _mn
111
(x
)11=
x 11111
(2 , 3)3 Z 23
3 .33
3
my)!” =
: xlllylll
xmym
(xy)"'
I
"I
(2 · 3) 3
Xm
(~)m
(y) =z ;::
w
2
,
'1
W1
7 xA_,-mIn =
xm
7
3
·3
23
(~)3
(g) =z ~:
3‐3
33 -‐44
1
=2
1
_
g
_ 2_
=
34
CHAPTER 1 EXPONENTS & RADICALS
Many students don 't know the difference between
( - 3) 2 and - 32
Order of operations (PEMDAS) dictates that parentheses take precedence . So,
( - 3) 2
= (-
3) · (- 3)
=9
Without parentheses, exponents take precedence:
- 32 = - 3 · 3 = - 9
The negative is not applied until the exponent operation is carried through. Make sure you understand this so
you don't make this common mistake . Sometimes, the result turns out to be the same, as in :
( - 2) 3 and - 23
Make sure you see why they yield the same result.
EXERCISE1: Evaluate WITHOUT a calculator . Answers for this chapter start on page 272.
(- 1)4
1.
l. t‐1)4
3)3
10. -(‐(‐3)3
50
19. 50
(- 1)5
2
2. (‐1)5
11. -(- 6)2
2
20. 332
( - 1) 10
3.
3. (‐])lo
12. - (- 4)3
3‐22
21. 3-
- 1) 15
4. ((21)15
4.
13. 23 X 32
(- 1)8
5.(‐-1)8
5.
14. ( - 1)4
- 18
6.
6. ‐18
15. (- 2)3 X (- 3 )4
2
24. 772
7.-‐(‐1)8
7. - ( - 1)8
16. 30
7- 2
25. 7‐2
s.(‐3)3
8. ( - 3)3
17. 6- 1
3
103
26. 10
9.-‐33
9. - 33
18. 4- l
27. 1010‐33
(- 1)5
22. 53
53
33 X 22
23. 5‐3
5- 3
X
X
8
PANDA
THE COLLEGE
COLLEGE PANDA
THE
The
calculator. The
a calculator.
use a
NOT use
exponents. Do NOT
positive exponents.
contains only
answer contains
your answer
that your
so that
Simplify so
EXERCISE
EXERCISE 2: Simplify
only positive
272.
page
on
start
chapter
this
for
Answers
you.
for
done
been done
have been
first ttwo
w o have
you. Answers
this chapter start
page
1.
=6x5
6x 5
3x 2 · 2x 3 =
1. 3x2-2x3
i
21.
21,
6,‐{1
6u 4
8112
'8uz
2.
2k‐44 ~4k2
: %
· 4k2 =
2. 2k-
12
12.
k
5x4 · 3x -‐22
3.
3” 5x -3x
14.
14.
(2x 2)- 3
5. (23r2)’3
2 a
(mnz)2
x22
X7.
x -3
1
23. %
23x- 2
I l
x
mn
4
3x
15_ W3x4
15.
(x-2)2
b-‐ 3 .• 3a6. -_ 3a
3a2b
3a ‐ 5b8
3n7
3117
7- 55
6n 3
7.
24. flm2n3
k- 2
k‐z
25.
25- pk-3
3
x1
xi?
16. 7 I
xi
x7
(a2b3)2
8. (azb3)2
26.
m2 3
(7)
c:
n:r
27
27.
(
17.
17 x2.
x2 x3.
‘r1 x4
x4
W4
9. ( xy4 )
x3y2
9,(”fl/2)
‘
2)- 3 • 2x 3
18. (x
(.r2)'3-2x3
3
(- x) 3
10.' -_(_-")
(b- 2) -3. (b3)2
(b‘2)‐3
, (b3)2
)3
(m2n
("1 fl)‘
22. ‐‑(mn 2)2
22.
- 4u 2v
2u v 2 • ~4uzv
13. 21402-
-3
- 3111
7m 3 •-‐3m‘3
4.
4. 7m3
10
a- 2)2
a- 1 •~a‘2)2
20. ((a‐1
(x2y - 1)3
11.
11. (x2y_1)3
3 )2
) 2 •- (3111
19. (2m
(2m)2
(3m3)2
2a
24 )
x y~z4
x2y3
4z-5
y
-3
x
x‘3y'42“5
?
off y ?
terms o
the value
what is the
then what
3x+2 = y, then
EXAMPLE
If3H2
value ooff 33x
j r in terms
EXAMPLE 1: If
A
)y+9
A)y
B
) y-- 9
B)y
ql
C)%
3
D)%
only difference
the exponent
the 2 in the
that the
notice that
we notice
Here we
what x is. Here
finding what
trouble of finding
Let’s
exponent is the
the only
difference
the trouble
avoid the
Let's avoid
out:
2
the
extract
let's
exponents,
laws
our
using
So
want.
we
what
between
the
given
equation
and
what
we
want.
50
using
o
u
r
laws
of
exponents,
let’s
extract
the
out:
and
equation
given
the
between
3X+2:3X.32=y
= 3x . 32 = .1/
3x+2
3xx=: 'i
3
9
Answer
Answer ~(D) .
a?7
value of a
the value
what is the
ra +7 , what
= 3‐“7,
3a+l =
ff 3"+1
EXAMPLE 2: If
EXAMPLEz-
equal.
be equal.
therefore be
muu sstt therefore
the same.
are the
bases are
the bases
Here
same. The
The exponents
exponents m
that the
Here we see that
a + 1l ==-~ aa+
+7
a+
2
2aa=: 6
au=@J
=l
9
CHAPTER
CHAPTER 1 EXPONENTS
EXPONENTS & RADICALS
RADICALS
a
4a
EXAMPLE
3: U 2a - b = 4, what is the value of b ??
EXAMPLE3zlf2a‐b=4,whatisthevalueof§5
2
2.
Realize
Realize that
that 4 is just
just 222.
(22V 22"
4 2.
?:_2h_=F:2
:2
4"
Za‐b
Square
Square roots
roots are
are just
just fractional
fractional exponents:
exponents:
I
.Jx
xi =
x?l
But what
i ? The
top means
bottom means
means to cube
cube root
root it:
what about
about xxi?
The 2 on
on top
means to square
square x. The
The 3 on the bottom
3/x2
We
this more
Wecan
can see this
m o r e clearly
clearly if we
we break
break it down
down::
g
l
2
2 I
3;-;,
2 (x
(x2)?
= {7x2
xx33 =
)1 =
v x2
The order
-rooting doesn’t
doe sn't matter
order in which
which we
we do
do the
the squaring
squaring and
and the
the cube
cube‐rooting
matter..
2
x? = (xhz : (3/32
The
result just
the cube
the outside.
outside . That
That way, we
don't need
need the
the parentheses.
parentheses.
The end
end result
just looks
looks prettier
prettier with
with the
cube root
root on
on the
we don’t
EXAMPLE
EXAMPLE 4: Which
Which of the
the following
following is
is equal
equal to ~
Vac‐5 ?
?
A)
A)x
B)
.x5- x4
B)x5‐x4
CM;
on?
1
The
fourth root
fractional exponent
The fourth
root equates
equates to aa fractional
exponent of 3', so
so
4
4
x5=x
Answer §).
(C) .
Answer
10
10
PANDA
COLLEGE PANDA
THE COLLEGE
THE
root, factor
square root,
a square
"surds"). To simplify
called ”surds").
square roots
simplifying square
on simplifying
you on
also test you
SAT will also
The SAT
roots (also called
simplify a
pairs :
any pairs:
out any
and take out
root and
square root
inside the square
number inside
the number
the
JT:\/2~2.2~z= \J /[I1]-[I}]
- - ' 3 = 2 --32=\2·2v3
/ § = 4= \4v3
/§
v'48 = ✓2-2·2 · 2 · 3 =
[I}] .
pair -.
second pair
out for the second
another 2 out
Then we take another
[I}] . Then
the first -.
out for the
a 22 out
take a
we take
above, we
example above,
In the example
looked
have
would
route
quicker
a
course,
Of
4.
get
to
root
square
the
outside
2's
two
the
multiply
we multiply
Finally, we
t w o 2’s outside
square root
course, a quicker route would have looked
this :
like this:
\/4‐
= t✓~/ _ - 3 =: 4\/§
4v3
v'48 =
example:
another example:
Here's another
Here's
M: \/--~2=2-3\/§:6\/§
pair:
as a pair:
root asa
square root
under the square
back under
put itit back
and put
outside and
number outside
the number
take the
backwards, take
To go backwards,
To
N
\/6-6- =
V72
= x/fi
6v'2i =: ~
any triplets
out any
ifi6, take
as W,
cube root such
a cube
simplify a
To simplify
such as
take out
triplets::
W=x7-2=2¢/§
0?
where x > O?
¾,
(x2 ) Q
form of (12)
equivalent form
the following
Which of the
EXAMPLE 5: Which
following is an
anequivalent
, where
A)..jx
AWE
CW?
x..fi
B)
Bm/E
DN/E
Solution
1:
Solution 1:
3
s
1
(x2)“ zxz4 ZYZ
zmz
--x=x\/§
Answer
Answer §)(B) .
3
.
.
73
23
2
. exponent of 3 to the exponent of x in
, each of the
compare this
(x2 ) 4 z= x 44 =
Solution 2: Since (x‘)El
Solution
z x 2- ,, we
we can
can compare
exponent 5
exponent
each the
2,1
3
3
2
answer choices.
choices.
answer
I
,Ix== xx2
Choice
Choice A:A: J?
1
I
1 I- iI
5
3
ChoiceB:
Choice 8:
= xx21
x\/§=x‘-xz
= x 1 3
x,/x = x1 • x 2 =x”Z
Choice
Choice C:
W
Choice
Choice D:
é/E
: xxI1
-1/x=
2
= x‘1
1
I
that the answer
confirm that
results confirm
These
answer is B.
These results
11
& RADICALS
CHAPTER 1 EXPONENTS &
RADICALS
EXERCISE 3: Simplify the radicals or solve for x. Do NOT use a calculator. Answers for this chapter start on
page 272.
page272.
1. \/1‐2
l.
Ju
/128
' 10. v128
2.
2. fi)%
11
ll.Sv'2
= ./x
3. v'45
1/5
3.
12.
12 3,/x
4. \/1_8
/18
13 2v'2= v'4x
13.
5.
5 2/fl
2m
14.
144)6
6
3\/7_5
6. 3/75
15
7. «3‐2
7.J32
16.
16 4/3x
8. m
v'200
17.
17 3v8
= x/2
9.
9. x/é
v8
18.
18 x,/x
= v'2l6
12
12
= v'45
= 2/3x
= 2,/6
THE
COLLEGE PANDA
PANDA
THE COLLEGE
CHAPTER EXERCISE:Answers for this chapter start on page 272.
A calculator should
NOT
be used
should N
O T be
used on the
following
following questions.
questions.
If
__
1
~
xc for all positive
positive values
what is
\/x‐
= x‘
values of xx,, what
V}
3
yX
the value
value of c ??
the
I
If aa- 1f =
what is the
= 3,
3, what
the value
value of a?
a?
A)
A) - 9
1
B) 9
1
C) 3
D)
D) 9
If 3x
3x =
= 10,
what is the
lu e of 3xIf
10, what
the va
value
3"’33 ??
10
A)
~
3
A ) yy ++ 3
A)
B)
39
B)) yy-‐ 3
B
C)
C ) 33- ‐ yy
D) 3y
3y
10
10
C) 57
27
2xx
IfIf 27
Y =
2
3, then
then x must
must equal
equal
223,
10
10
27
27
D) fi
10
10, what
what is the
the value
value of y2
y °?
If y3/55 = 10,
IfIf a
are positive
integers, which
a and
and b
bare
positive even
even integers,
which of the
the
following
following is
is greatest?
greates t?
A ) 440
0
A)
B)
400
B) 400
A)
A)
8)
B)
C)
C)
C) 1,000
1,000
D) 10,000
(- 2a)b
(‐2a)"
( - 2a) 2b
<‐2a)2"
(2a) 11
an)”
D) 2112"
2a 2/J
D)
ifii't,
The expression
expression Wx2y4, where
where x > 00 and
and y > 0,
O,is
is
The
equivalent
which of the
the following?
following?
equivalent to which
A)
A)
32 =
9?
If x2
3/3, for what
what value
= yy9
?
x2 =
= y3,
value of 2
z does
does xX32
PY
x/Ty
A
A)) -‐ 1
B)
B ) y ,fi
y fi
B) 0
1
C) x2
C) 1
D) x 2y
D) 2
13
13
CHAPTER 1 EXPONENTS
CHAPTER
EXPONENTS &
& RADICALS
RADICALS
If
jxJx = xa, then
If 2”3
2x+3 -‐ 2X=
2x = k(2x), what
If
what is the
the value
value of k
k??
what is the value of a?
1
A) 2
A
A)) 3
3
C) 7
B) 5
B) B)
4
0) 8
D)
C) 1
C)
4
D)
0) 3
A calculator is allowed
allowed on the following
following
questions.
questions.
lH
30, x > 1,
lf xac • xhc = xx30,x
lfx“‘oxh‘
1 ,and
a n daa+
+b =
= 55,, wwhat
h a t iiss
1
3 4
24
Jf(S 3) 4k:
k = (53
(5! ))24,, what
what is
is the
the value
value of
of k
k??
the value
value of c?
the
c?
A) - 6
A) 3
2
B) 5
B) 3
C) 6
3
C) 4
0)) 110
D
0
0) 2
If 4
2n+ 3
.
?
= 811+ 5, what
what is
1s the
the value
value of
of n
11 ?
2n
Which of the
the following
Which
following is equivalent
equivalent to
to x7?
x -;; for
for
all
b
are
all positive
positive values
values of
of x, where
where a and
and bare
positive integers?
positive
integers?
A) 6
B) 7
C
C)) 8
A) vbax2
lftix2
D
0)) 9
½2n
B) x7x2“
C)
tJxa 12
C) (7x1142
D)
0)
5
w,
2"xb
X
Which of the
Which
?
the following
following is equivalent
equivalent to
to ((‐2)3
?
2) 3
A)
A) -‐2-\3/Z
2-¼
B)
B) 22· 3/1
</4
C)
C) -‐4-€/i
4· '7'2
If x2y3
x 2y 3 =
= 10
10 and
and x3y2
x 3y 2 =
If
= 8, what
what is the
the value
value of
x5y5?
x5y5 ?
D)
3/2
0) 4.
4 · v'2
A
8
A)) 118
B
0
B)) 220
C
C)) 4400
D
0)) 8800
14
14
Per
cent
Percent
EXAMPLE
the questions
EXAMPLE 1: Jacob got
got 50% of the
correct on
30-question test
test.
questions correct
on a
a 30-question
test and
and 90%
90% on
on a
a 50
50 question
question test.
What
What percent
percent of all questions
Jacob
get
correct?
questions did
did Jacob get correct?
First, let's
First,
let’s find the total
total number
number of questions
questions he
he got
got correct:
correct :
50%
50%><30:
X 30 =
1
2 xX 3300 =: 1155
9
90/ox507mx50‐45
90% X 50 = 1Q X 50 = 45
0 : Z : -.
So
15+
z 60
u t of
50 =
; 80
So he
he got 15
+ 45 =
60 questions
questions correct
correct oout
of a
a total
total of
of 30
30 +
+ 50
80 questions:
questions : 2‐0
~~ =
~ = I 75% I.
EXAMPLE
2: The
price of
by 20%,
20%, then
decreased by
by 40%,
40%, then
by 25%.
25%. The
The
EXAMPLE 2:
The price
of a
a dress
dress is
is increased
increased by
then decreased
then increased
increased by
final price
is what
what percent
percent of
of the
the original
price?
final
price is
original price?
Here’s
technique for dealing
dealing with
“series of
of percent
Let the
Here's the technique
with these
these "series
percent change”
change" questions.
questions . Let
the original
original price
price be
be p.
p.
When
you
multiply
by
1.20
because
it’s
the
original
price
plus
20°/o.
When
it’s
decreased
When ppisis increased
increased by 20“/o,
20%, you multiply by 1.20 because it's the original price plus 20%. When it's decreased
by
is what's
what’s left
left after
O u r final
by 40%, you
you multiply
multiply by
by .60 because
because 60%
60% is
after you
you take
take away
away 40°/o.
40%. Our
final price
price is
is then
then
p
.90p
p Xx1.20
1.20 X><.60 x
X 1.25
1.25 := .90p
The final price
price is
is I90"/o
90% Iof
of the original
original price.
price.
Example
IMPORTANT percent
concept by
calculate
prices
Example 2 shows
shows the MOST IMPORTANT
percent concept
by far
far on
on the
the SAT.
SAT. Never
Never ever
ever calcu
late the
the prices
at each
to get
get the
the end
end result.
each step.
step. String
String all the changes
changes together
together to
result .
It’s
important to know
know why
why this
works. Imagine
Imagine again
want to
increase it
it by
by
It's important
this works.
again that
that the
the original
original price
price is
is p
p and
and we
we want
to increase
20"o.
Normally,, we
just take
p and
and add
add 20%
20% of
of it
it on
top:
20%. Normally
we would
would just
take p
on top :
p
+ .20p
p+
.20p
But realize
realize that
that
p
+ .20p
.20p == pp(1+.20)
:1.20p
p+
( l + .20) =
l.20p
15
CHAPTER
CHAPTER 2 PERCENT
PERCENT
And
And now
n o w we want
want to decrease
decrease this
this new
n e w price
price by 40%:
1.20p -‐ (.40)(
(.40)(l.20p)
: (1.20p)(
(l.20p)(ll -‐ .40)
.40) := (1.20p)(.60)
z (l.20
(1.20)(.60)p
l.20p
1.20p ) =
(1.20p)( .60) =
)( .60)p
which proves
proves we
we can
can calculate
calculate the
the final
final price
price directly
directly by
by using
N o w we
re set
set up
up to
tackle the
the
which
using this
this technique.
technique. Now
we're
to tackle
inevitable
compound
interest
questions
on
the
SAT.
inevitable compound interest questions on the
EXAMPLE 3: Jonas
Jonas has
has aa savings
savings account
account that
that earns
earns 33 percent
percent interest
annually. His
His initial
initial
EXAMPt:.E
interest compounded
compounded annually.
deposit was
was $1-000.
$1000. Which
Which of
of the
the following
following expressions
expressions gives
value of
10years?
deposit
gives the
the value
of the
the account
account after
after 10
years?
A) 100Q(l,30)l0
A)1000(1.30)1°
1000+30(10)
B) 1000
+ 30(10)
C) 1000(1.03)
1000(1.03)(10)
C)
(10)
10
D) 1000(1.03)
1000(1.03)10
A 33 percent
interest rate
rate compounded
compounded annually
annually means
means he
he earns
on the
year. Keep
Keep in
A
percent interest
earns 33 percent
percent on
the account
account once
once a
a year.
mind that
that this
isn't just
just 3%
3% on
on the
the original
original amount
amount of $1000. This
3%of
at the
time,
mind
this isn't
This is 3%
of whatever’s
whatever's in the
the account
account at
the time,
including any
any interest
interest that
that he's
he's already
already earned
earned in
in previous
This is the
the meaning
meaning of
of compound
compound interest.
interest.
including
previous years.
years. This
80 ifif we're
we’re in
in year
year 5,
5, he
he would
3°/o on
on the
the original
original $1000 and
interest deposited
deposited in years
years 11
So
would earn
earn 3%
and 3°/o
3% on
on the
the total
total interest
through
through 4.
4.
If we
we try
try to calculate
calculate the
the total
after each
each and
and every
every year,
year, this
this problem
forever. Let's
Let’s take
what we
we
If
total after
problem would
would take
take forever.
take what
learned from
Example 3 and
here:
learned
from Example
and apply
apply it here:
total:: 1000(1.03)
1000(1.03)‐
‐ 1000(1.03)
1000(1.03)11
=
Year 11 total
Year
2 total
1.03) = 1000(1.03)
Year2
total:: 1000(1.03)(
1000(1.03)(1.03)‐_
1000003)22
Year
3 total:
(1.03)( 1.03)( 1.03) = 1000(1.03)
Year3
total: 1000
1000(1.03)(1..03)(103)
1000003)33
Year
4 total:
1.03)( 1.03)(
1.03) = 1000(1.03) 4
Year4
total: 1000(1.03)(
1000(1.03)(1.(o3)
103)(1.03)=1000(1.03)4
See the
whatever the
was last
last year.
the pattern?
pattern? Each
Each year
year is an
an increase
increase of 3%
3°/o so
so it's
it’s just
just 1.03 times
times whatever
the value
value was
year. Note
Note
ten times
that
're not
price of a
a dress
dress being
that we
we’re
n o t doing
doing any
any calculations
calculations out.
out. Think
Think of it as
as the
the price
being increased
increased by 3%
3°/o ten
times..
§J .
10
, answer
Therefore,
the Year 10
Therefore, the
10total
total is 1000(1.03)
1000(l.03)1°,
answer ( D ) .
Most
the equation
equation A
Most of these
these compound
compound interest
interest questions
questions can
can be
be modeled
modeled by the
A=
= P(l
P(1+ r)t,
r)‘, where
where A
A is
the
amount accumulated,
accumulated, P is the
initial amount,
the total
total amount
the principal
principal or the
the initial
amount, r .is
is the
the -interest
interest rate,
rate, and
and t is the
the
number of times
times interest
interest is received.
received.
number
EXAMPLE
account that
that earns
earns 5 percent
each year,
EXAMPLE 4: Jay puts
puts an
an initial
initial deposit
deposit of $400 into
into aa bank
bank account
percent interest
interest each
year,
compounded
annually.
Which
of
the
following
equations
gives
the
total
dollar
amount,
A,
in
the
account
compounded annually. Which the following equations gives the total dollar amount,
the account
after
years?
after t years?
A) A = 400(1.0St)
400(1.05t)
A)
B) A =
= 400(0.0St)
4oo(o.051)
C)
C) A
A = 400(0.0Sl
4oo(o.05)'
400(1.05) 1
D) A =
= 400(1.05)'
After
and the
the initiaJ
amount P is 400. Plugging
After t years,
years, interest
interest has
has been
been received
received t times.
times. The
The rate
rate r is 0.05 and
initial amount
Plugging
§J .
these
into the
these values
values into
the formula,
formula, we
we see
see that
that the
the answer
answer is ( D ) .
16
THE
COLLEGE PANDA
THE COLLEGE
PANDA
EXAMPLE
less eggs
eggs than
they did
did last
year. If
If they
they laid
laid 3,500
EXAMPLE 5: This year,
year, the
the chickens
chickens on
on aa farm
farm laid
laid 30% less
than they
last year.
eggs
eggs this year,
year, how
how many
many did
did they
they lay
lay ~t
last year?
year?
This Year =
(Last Year)
: (.70)
(.70)(Last
3,500
Last Year)
3,500 =
: (.70)(
(.70)(Last
I5,0001 =
Last Year
Last
Percent
calculated as
as follows:
Percent change
change (ak.a.
(a.k.a. percent
percent increase/decrease)
increase/ decrease) is calculated
follows:
1 value
al
_ new value -‐ old
ch
°/o
,ochange
ange -=
01
n_______________ew
vaollude
“13: v ue x 100
oId v alue
For
and rises
dollars , the
the percent
percent change
For example,
example, if
if the
the price
price of aa dress
dress starts
starts out
out at
at 80
80 dollars
dollars and
rises to 90 dollars,
change is:
90
9 0 ‐ 880
0
12.5%
;80 Xx 100 = 12.5%
percent change
change is positive,
it’s aa percent
percent increase.
increase. Negative?
Negative? Percent
It’s important
important to remember
remember
If percent
positive, it's
Percent decrease.
decrease. It's
that
based on
that percent
percent change
change is
is always
always based
on the
the original
original value.
value.
EXAMPLE 6: In aparticular
store, the
the number
number of TVs sold
sold the
the week
Friday was
was 685. The
The number
number
EXAMPLE
a particular store,
week of Black Friday
of TVs sold
sold the
the following
following week
week was
was 500. TV
TV sales
sales the
the week
week following
following Black Friday
Friday were
were what
what percent
percent less
less
of
sales the
the week
week of
of Black
Black Friday
Friday (rounded
(rounded to
to the
the nearest
nearest percent)?
percent)?
than TV sales
than
A)17%
A) 17%
B)27%
B) 27%
C)
37%
C)37%
D) 47%
0)47%
W ~ ‐0.27
500 -‐ 685
685 ~ - 0 27
500
685
.
We put
the difference
difference over
over 685, NOT
N O T 500. Answer
Answer ~(B) -.
put the
We
EXAMPLE 7:
7: In
In aaparticular
store, the
the number
number of
of computers
computers sold
the week
Black Friday
Friday w.as
was 470.
470. The
The
EXAMPLE
particular store,
sold the
week of
of Black
number
of
computers
sold
the
previous
week
was
320.
Which
of
the
following
best
approximates
number of computers sold the previous week was 3ZO. Which of the following best approximates the
the
percent increase
increase in computer
sales from
from the
the previous
previous week
week to the
week of Black Friday?
Friday?
percent
computer sales
the week
A) 17°/o
A)17%
B) 27°/o
8)27%
C) 37%
C)37%
D) 47°/o
0)47%
320 ~ O
470 -‐ 320
a: 0.47
47
320
·
This time,
time, the week
week of
of Black Friday
Friday is
n o t the
the "original"
”original” basis
for the
percent change.
Weput
put the
is not
basis for
the percent
change. We
the difference
difference
over the previous
previous week's
week’s number,
number, 320. The answer
answer is ~(D) -.
over
17
17
PERCENT
CHAPTER
CHAPTER 2 PERCENT
percent:
involving percent:
examples involving
more examples
A few more
oi
number of
the number
school decreased
number of students
EXAMPLE 8: The number
EXAMPLE
students at aa school
decreased 20% from
from 2010 to 2011. IfIf the
2010
in
enrolled
students
of
number
the
expresses the number students enrolled in
the following
which of the
2011
enrolled in 201
students enrolled
students
1 was
was k, which
following expresses
?
terms of kk ?
in terms
A) 0.75k
A)
B) 1.20k
1.20k
C) 1.25k
1.25k
D)
D) 1.Sk
1.5k
value .
the new
and not
(from 2010) and
value (from
original value
the original
is based
change is
Percent change
NOT 1.20k. Percent
is NOT
answer is
The answer
The
based off of the
n o t the
n e w value.
students in 2010,
number of students
the number
be the
Let x be
.80x =
: k
.80x
x=
z 1.25k
1.25k
Answer (@
more students
were 25% more
there were
Therefore, there
Therefore,
students in 2010 than
than in 2011. Answer
(C) .
those Red
Among those
are Red
students are
a school,
at a
graders at
10th graders
Among 10th
EXAMPLE 9: Among
EXAMPLE
school, 40% of the
the students
Red Sox fans. Among
Red
Sox fans
both Red
are both
the school
graders at the
10th graders
What percent
fans. What
are also Celtics fans.
Sox fans,
fans, 20% are
percent of the
the 10th
school are
Red Sox
fans
and
fans?
and Celtics fans?
it's 100.
that it's
suppose that
let's suppose
number of 10th
the number
know the
don't know
We don’t
We
10th graders
graders at
at the
the school
school so
so let’s
= 40
= 40'3/o
Red
40% of 100 =
fans =
Red Sox fans
=8
40 =
Celtics
fans =
: 20% of 40
Red Sox fans
Celtics & Red
The
= I8% I
then %~ =
answer is then
The answer
1
typically 100.
total, typically
the total,
represent the
number to represent
up aa number
questions is to make
percent questions
A ccommon
o m m o n strategy
strategy in percent
make up
18
18
THE COLLEGE
COLLEGE PANDA
PANDA
THE
CHAPTER EXERCISE:Answers for this chapter start on page 276.
A calculator is allowed
allowed on the following
following
questions.
questions.
If x is 50% larger
than 2,
z, and
and y is
than
If
larger than
is 20% larger
larger than
z, then
then x
xisis what
what percent
percent larger
larger than
than y
y??
2,
A) 15%
Reid
purchase aa rug
Reid wants
wants to purchase
rug that
that has
has aa price
price of
$150.00. He has
ha s aa coupon
reduce the
coupon that
that would
would reduce
the
cost
cost of the
the rug
rug by
by k%. If
If the
the coupon
coupon would
would
mg by $12.75, what
reduce
the cost
reduce the
cost of the
the rug
what is the
the
value
value of k?
k?
B) 20%
C) 25%
D) 30%
Veronica has
has aa bank
account that
earns m%
Veronica
bank account
that earns
m%
interest compounded
compounded annually.
annually. If she
opened the
interest
she opened
the
account with
with $200, the
the expression
expression $200(x)‘
$200 (x) 1
account
represents the
the amount
amount in the
the account
represents
account after
after t
years. Which
Which of the
the following
gives x in terms
terms
years.
following gives
m??
of m
A) 1 +
+ .01m
.Olm
In March, a city zoo attracted 32,000 visitors to
its polar bear exhibit. In April, the number of
visitors to the exhibit increased by 15%. How
many visitors did the zoo attract to its polar bear
exhibit in April?
B) l1 +
+m
m
C) 1 -‐ m
D)
D) 1 ++ 100m
100m
A)
A) 32,150
B) 32,480
C) 35,200
A charity organization collected 2,140 donations
last month . With the help of 50 additional
vo lunteer s, the organization collected 2,690
donations this month. To the nearest tenth of a
percent, what was the percent increase in the
number of donations the charity organization
collected?
D) 36,800
Miguel is following a recipe for marinara sauce
that require s half a tablespoon of vinegar. If one
cup is equivalent to 16 tablespoons,
approximately what percent of a cup of vinegar
is the amount required by the recipe?
A)
A) 20.40/o
20.4%
B) 20.7%
C) 25.4%
A) 2.3%
D)
D) 25.7%
B) 3.1%
3.10/o
C) 9.4%
D) 12.5%
12.5%
19
19
CHAPTER 2 PERCENT
The number
number of dishes
served by a restaurant
The
dishes served
restaurant
during dinner
dinner was
was 17.5% greater
than the
the
during
greater than
If the
number of dishes
dishes served
served during
during lunch.
lunch. If
number
the
restaurant served
served 940 dishes
dishes during
during dinner,
restaurant
dinner, how
how
many
more
dishes
did
the
restaurant
serve
many more dishes did the restaurant serve
during dinner
dinner than
than during
during lunch?
during
lunch?
The discount price of a book is 20% less than the
retail price. James manages to purchase the book
at 30% off the discount price at a special book
sale. What percent of the retail price did James
pay?
A) 42%
8)
B) 48%
4896
C) 50%
5095
D)
[)) 56%
5696
Each
pistachios left
Each day,
day, Robert
Robert eats
eats 40% of the
the pistachios
in his
his jar at that
time . At the
the second
that time.
the end
end of the
second
day, 27
pistachios remain
day,
27pistachios
remain.. How
How many
many pistachios
pistachios
were in the
were
the jar at the
the start
start of the
the first day?
day?
ln 2010, the
the number
number of houses
built in Town
in
houses built
Town A
was 25 percent
percent greater
than the
the number
was
greater than
number of
houses
built in Town
Town B.
8. If
houses built
If 70 houses
houses were
were built
built in
Town A during
during 2010, how
many were
were built
built in
Town
how many
Town B?
8?
Town
A)
A ) 775
5
8)
80
B) 80
C)) 885
C
5
D)
D ) 995
5
Joanne bought a doll at a 10 percent discount off
the original price of $105.82. However, she had
to pay a sales tax of x% on the discounted price .
If the total amount she paid for the doll was
$100, what is the value of x?
Over a
a ttwo
week span,
span, John
John ate
20 pounds of
Over
w o week
ate 20pounds
chicken
and 15
15 pounds
dogs.. Kyle
chicken wings
wings and
pounds of hot
hot dogs
ate
ate 20 percent
percent more
more chicken
chicken wings
wings and
and 40
percent more
more hot
Considering only
only
hot dogs
dogs.. Considering
percent
chicken
dogs, Kyle ate
ate
chicken wings
wings and
and hot
hot dogs,
approximately
weight,
percent more
more food,
food, by weight,
approximately x percent
than
John.
What
is
x
(rounded
to
the
nearest
(rounded
the nearest
than John. What
percent)?
percent)?
A) 2
B)
8) 3
C) 4
D) 5
A ) 225
5
A)
B)
8) 27
27
C)) 229
C
9
D
0
D)) 330
20
20
PANDA
COLLEGE PANDA
THE COLLEGE
ll)
that
bond that
government bond
Omar
holds a government
currently holds
Omar currently
year, the
Each year,
value of $900. Each
market value
a market
has a
has
20%
be 20°/o
expected to be
bond is expected
the bond
value of the
market value
market
If
before. If
year before.
the year
value the
market value
than its market
higher than
higher
constant,
a constant,
pisis a
where p
+ p), where
expression 900(1 +
the expression
the
the
value of the
market value
expected market
the expected
represents the
represents
?
p
value of ?
what is the value
years, what
after 3 years,
bond after
bond
Jane is playing a board game in which she must
collect as many cards as possible. On her first
tum, she loses 18 percent of her cards. On the
second tum, she increases her card count by 36
percent. lf her final card count after these two
turns is 11, which of the following represents her
starting card cow1t in terms of n ?
A)
A)
n
11
(1.18)(0.64)
(l.18 )(0.64)
(l.18 )(0.64)n
B)
8) (1.18)(0.64)n
C)
n
C) (1.36)(0.82)
(0.82)( 1.36)11
D) (0.82)(1.36)n
groceries in 2015. She
on groceries
dollars on
spent xx dollars
Sims spent
than in
groceries in 2016 than
on groceries
more on
spent
34% more
spent 34°/o
groceries in
more on groceries
145% more
spent 145°/o
she spent
2015, and
and she
following
Which of the following
2017 than
than in 2016. Which
dollars,
amount, in dollars,
the amount,
represents the
expressions
expressions represents
Sims spent
groceries in 2017?
spent on groceries
Due to deforestation , researchers expect the deer
population to decline by 6 percent every year. If
the current deer population is 12,000, what is the
approximate expected population size 10 years
from now?
A)
B)
8)
C)
C)
D)
A) 4800
B)
8) 6460
C) 7240
D)
D) 7980
(2.45)(0.34x)
(2.45)(0.34x)
(1.45)(0.34x)
(1.45)( 0.34x)
(2.45)(1.34x)
(2.45)( 1.34x)
(1.45)(1.34x)
(1.45) (1.34x)
In 2016, County
the
collected the
8 collected
County B
and County
County A and
same
amount of
the amount
ln 2017, the
taxes . In
amount of taxes.
same amount
taxes collected
decreased by 25%
County A decreased
collected by County
and
8
County B
collected by County
taxes collected
amount of taxes
the amount
and the
increased
by
20%.
If
County
A
collected
60
Cow1ty collected
increased
million
the
was the
what was
taxes in 2017, what
dollars of taxes
million dollars
amount
taxes, in millions
8
County B
dollars, County
millions of dollars,
amow1t of taxes,
types of
A small
different types
store sells 3 different
clothing store
small clothing
accessories:
and
ties, and
are ties,
60% are
scarves , 60°/o
are scarves,
accessories: 20% are
the
accessories are
the
half of the
belts. If half
are belts.
other 40 accessories
the other
ties are
with scarves,
many scarves
scarves
how many
scarves, how
replaced with
are replaced
will the store
have?
store have?
collected
collected in 2017?
A
4
A)) 554
B)
78
8) 78
C
0
C)) 990
D
6
D)) 996
21
21
CHAPTER 2
M
CHAPTER
PERCENT
PERCENT
Daniel has
Daniel
has $1000 in
in a
checking account
account and
a checking
and
$3000 in a savings
savings account.
account. The checking
checking
account
account earns
earns him
him 1 percent
percent interest
interest
compounded annually
compounded
annually.. The savings
savings account
account
earns him
him 6 percent
earns
percent interest
interest compounded
compounded
annually
annually.. Assuming
Assuming he
he leaves
leaves both
both these
these
accounts
alone, which
accounts alone,
of the
the following
which of
following
represents how
represents
much more
how much
more interest
interest Daniel
Daniel will
will
have
have earned
earned from the savings
savings account
account than
than from
from
the checking
checking account
account after
after 55 years?
years?
:
*
,
A) 3,
3, 000(1.06
OOO(1.06)5
1,000(1.01)5
)5 ‐- 1,
000(1.01)5
B) 3,000
(1.06)( 5) -‐1,000(1.01)(5)
B)
3,000(1.06)(5)
1,000(1.01)(5)
5
C) (3,000
(1.06) -‐ 3,000
C)
(3,000(1.06)5
3,000)) -‐ (1,000(1.0
(1,000(1.01)5
1)5 -‑
1,
000)
1,000)
D)
000(1.06) (5) - 3,000 ) D) (3,
(3,000(1.06)(5)‐3,000)‑
(1,000(1.0
(1,000(1.01)(5)
1)(5) -‐ 1,000)
p ( 1 + 1~0)5
The expression
expression above
above gives
of
gives the population
population of
leopards
leopards after
after five years
years during
during which
which an
an initial
initial
population of P leopards
leopard s grew
grew by rr percent
population
each
percent each
year. Which of the following
following expressions
gives
expressions gives
the percent
percent increase
the
increase in the leopard
population
leopard population
over these five years?
over
years?
A)
A) (1(Hfi)
+ 1~)5
, .
B)
B)
(1 + 1~)5 - 1
(1+fir)
_:____!.=..:..-=-
x 100
‐1><100
(”m)
( 1 + 1~0)5
C)“ll‐fif‐llxloo
)5- 1] x 100
C) [ ( 1 + l ~O
D)(Ll-igafxloo
D) ( 1 + ~)5x 100
1
22
Exponential vs.
vs. Linear
Linear
Growth
Growth
The
account earning
The population
population of ants
ants doubling
doubling every
every month.
month. A bank
bank account
earning 55 percent
percent every
every year
year.. These
These are
are
examples
a quantity
quantity grows
examples of exponential
exponential growth,
growth, which
which occurs
occurs when
when a
grows periodically
periodically by aa factor
factor greater
greater than
than
account , it's
1.
1. In
In the
the case
case of the
the ants,
ants, this
this factor
factor is
is 2.
2. In
In the
the case
case of the
the bank
bank account,
it’s 1.05. When
When exponential
exponential growth
growth
happens,
happens, we
we can
can model
model it as
as an
an equation
equation that
that looks
looks like
like
y : ab,
where y is
is the
the final quantity
quantity after
after t time
time periods
periods (e.g.
(e.g. years),
years), a is the
quantity, and
and bis
b is aa growth
growth factor
factor
the initial
initial quantity,
where
greater
than
1.
Soif
we
started
off
with
100
ants,
o
u
r
model
equation
would
be
greater than 1. So if we started
with
ants, our model equation would be
: 100(2)
100(2)'1
y=
where tis
t is the
the number
number of months
months that
that have
have gone
gone by. And
And if
if oour
u r bank
bank account
account started
started off with
with $200, our
o u r equation
where
equation
would look like
like
would
: 200(1.0Sf
200(1.05)'
yy =
where tis
t is the
the number
number of years
years that
that have
have passed
passed.. You've
You’ve seen
already in the
the percent
chapter..
where
seen this
this already
percent chapter
Graphs of exponential
exponential growth
growth have
have the
the following
following shape:
shape:
Graphs
yll
Notice how
how the graph
graph creeps
creeps up
up slowly
slowly at
at first but
but then
then shoots
shoots up
faster over
over time.
time. That's
That’s exponential
Notice
up faster
faster and
and faster
exponential
growth.
growth.
23
CHAPTER
CHAPTER 3 EXPONENTIAL
EXPONENTIAL VS. LINEAR
LINEAR GROWTH
GROWTH
Exponential decay
decay,, however,
however, is the
Imagine aa radioactive
mass over
over time.
time. It
Exponential
the opposite.
opposite. Imagine
radioactive substance
substance that
that loses
loses mass
It
loses aa lot
lot of its
its mass
beginning and
and then
then loses
loses it more
more and
astime
loses
mass in the beginning
and more
more slowly
slowly as
time goes
goes by.
Mass
Mass
It’s worth
worth memorizing
memorizing the
the shapes
shapes of these
these graphs
graphs of exponential
exponential growth
The SAT may
may test you
you
It's
growth and
and decay. The
explicitly
explicitly on
on them
them..
The
exponential growth:
growth :
equation for exponential
exponential decay
decay is the
the same
same as
as the
the equation
equation for exponential
The equation
ab'1
y = ab
The only
only difference
So an equation
equation that
that models
exponential decay
difference is that
that the
the growth
growth factor, b, is less than
than 1. Soan
models exponential
decay
might
might look
look like
like
: 400(0.6
400(O.6)‘
y31=
)1
where
years from
that the
where y is the
the mass,
mass, in grams,
grams, of aa radioactive
radioactive substance
substance t years
from now.
now. The
The 400 indicates
indicates that
the substance
substance
currently has
has aa mass
400 grams
grams,, and
and the
the 0.6
0.6 indicates
that at
at the end
the substance
substance is left with
with
indicates that
end of each
each year, the
currently
mass of 400
mass it started
started the year
year with.
ln other
other words,
words, it loses
loses 40% of its mass
each year.
60% of the mass
with. 1n
mass each
Sofar, the
the examples
examples we've
we've discussed
discussed have
been relatively
relatively simple.
simple. To model
model more
more complicated
complicated cases of exponential
So
have been
exponential
growth and
and decay,
such as
asa
bank account
account growing
growing by 3%
3°/o every
substance losing
losing half
half
growth
decay, such
a bank
every 2 years
years or a
a radioactive
radioactive substance
its mass
every 9 months,
need to use
advanced exponential
exponential equation:
equation:
mass every
months, we'll
we’ll need
use aa slightly
slightly more
more advanced
I
yy=abi
= abl
where
Note that t and
and k
where k is
is the
the time
time required
required for y to increase
increase by one
one factor
factor of b.
17. Note
It must
must have
have the same
same
units. So
50 if
if t is in years, then
than k should also
also be in years.
years.
units.
Let's go
go over
some examples
so that
that you
you fully understand
understand what
Let's
over some
examples so
what k means.
means.
EXAMPLE 1:
EXAMPLE
= 400(1.05)
400(1.05)§!
M =
The equation
equation above
models the
the mass
mass M, in nanograms,
nanograms, of a
years. Based
Based on
the equation,
The
above models
a particle
particle after
after t years.
on the
equation,
which of the
following best
best describes
the mass
mass of the
the particle
particle over
which
the following
describes the
over time?
time?
A) It increases
increases by 5%
5% every
every 4 months.
months.
A)
increases by 5 nanograms
nanograms every
months.
C) It increases
every 4 months.
increases by 5%
5°/o every
B) It increases
every 3 years.
years.
increases by 5 nanograms
nanograms every
years.
D) It increases
every 3 years.
We have
have an
an exponential
exponential equation
with kk == 3
3 and
and bb == 1.05,
takes 3 years
the mass
mass of
We
equation with
1.05, which
which means
means it takes
years for the
the particle
particle to increase
increase by aa factor of 1.05. In other
other words,
words, the
years.
the particle's
particle 's mass
mass increases
increases by 5%
5% every
every 33 years.
Answer ~(B) .
Answer
24
24
PANDA
COLLEGE PANDA
THE COLLEGE
THE
EXAMPLE2:
EXAMPLEz‑
M = 400(0
400035)“
.6) 31
M
equation,
the equation,
on the
Based on
years. Based
after t years.
nanograms, of aa particle
mass M, in nanograms,
the mass
models the
above models
equation above
The equation
The
particle after
time?
over time?
particle over
the particle
mass of the
describes the
best describes
following best
the following
which of the
which
the mass
every 4 months.
It decreases
decreases by 60% every
months.
A) It
months.
every 4 months.
decreases by 40% every
C) ItIt decreases
by 60% every 3 years.
B) It decreases by
40%every 3 years.
D) It decreases by 40%
’' as a reference,
tt so how are
k'
form of k'
not
equation is n
given equation
exponent of the given
the exponent
that the
reference, we see that
ab1ias
Using y =
Using
= abI
o t in the form
so how are
arithmetic trick:
have to use
the value
supposed to find the
we supposed
we
value of k?
k? We
Wehave
use an
an arithmetic
3
400(0.6 /f( l / )
= 400(o.6)’/“/3’
400(0.6/ 1 =
M
M == 400(0.6)3’
1
1,
1
1
4
or 4
every 5 year, or
decreases by 11 ‐- 0.6 = 40% every
the particle's
means the
that k =
we can
Now
N
o w we
can see that
: 5, which
which means
particle’s mass
mass decreases
[ED.
Answer (C) .
months.
months. Answer
double
able to
organization is
the organization
If the
volunteers. If
50 volunteers.
has 50
currently has
organization currently
EXAMPLE
EXAMPLE 3: A non-profit
nOn-profit organization
is able
to double
of
number
the
odels
m
best
equations
following
the
of
which
months,
8
every
volunteers
of
number
the
the number volunteers every months, which the following equations best models the number of
now?
from
months
t
have
will
organization will have months from now?
the organization
v, the
volunteers, 0,
volunteers,
I
= 50(2)§
50(2)-S
A) v =
l
= 50(2)i
B) vU=
50(2)i
50(2)81
V = 509)!“
D) U=
50(2) 1
V = 50(2)t
C) v
1
1
number
= 2.
given information,
the given
on the
Based on
as a reference
abk asa
Again,
reference.. Based
information, a =
= 50
50 and
and b =
2. Since the
the number
use y := abl‘
we'll usey
Again, we’ll
equation
the equation
Therefore, the
months, k =
and t is in months,
months, and
every 8 months,
growth factor every
of volunteers
z 8. Therefore,
increases by the growth
volunteers increases
Answer ~(A) .~. Answer
volunteers is v = 50(2)
that
number of volunteers
50(2) 5.
models the number
best models
that best
double
organization is able
the organization
volunteers . H
has 50
currently has
EXAMPLE
50 volunteers.
If the
able to double
organization currently
non-profit organization
EXAMPLE4: A non-profit
the number
models the
equations best
following equations
the following
which of the
months, which
the
number of volunteers
volunteers every
best models
number of
every 8 months,
the number
now?
years from
will have
volunteers,
the organization
have t years
from now?
organization will
v, the
volunteers, 0,
t
A) v =
= 50(2)é
50(2)lJ
B) t)v =
50(2) 7f
= 50(2)%i
;i0(2) 81
= 50(2)8f
v=
D) 0
C) Vv =
= 50(2)
500)?¥
this, the
Because of this,
months. Because
instead of months.
years instead
except t is in years
one in Example
This question
same as
as the one
Example 3,
3, except
the
question is the same
convert
to
have
we
so
units,
same
the
have the same units, sowe have convert
must
and k m
the correct
answer is n
o t the
the same.
correct equation,
equation, t and
u s t have
same. To form the
not
answer
8
equation is then
= % =
gives k z
8 months
= ~5 years.
years. The equation
then
which gives
years, which
into years,
months into
12
1(2/ 3 ) = 50(2)37'
50(2) ¥
u:= 50(2/
sour/(W3)
V
[ED.
Answer
Answer (C) .
25
CHAPTER
EXPONENTIAL VS. LINEAR
CHAPTER 3 EXPONENTIAL
LINEAR GROWTH
GROWTH
Now
let's compare
compare exponential
exponential growth
growth and
and decay
decay with
with linear
linear growth
As you
know,
Now let's
growth and
and decay. As
you may
may already
already know,
linear
be modeled
example, if Ann
Ann has
has aa piggybank
linear growth
growth can
can be
modeled by
by aa line
line with
with aa positive
positive slope.
slope. For
For example,
piggybank with
with 100
dollars already
already in it,
it, and
and she
she adds
adds 5 dollars
dollars every
every month,
month, the
piggybank can
can be
be modeled
modeled
dollars
the total
total amount
amount in the piggybank
by
by
: St + 100
A =
where
the total
number of months,
(they-intercept)
initial amount.
where A is the
total amount,
amount, t is the number
months, and
and 100 (the
y-intercept) is the initial
amount.
A
Unlike exponential
consistent. There
There is no
Unlike
exponential growth,
growth, linear
linear growth
growth is constant
constant and
and consistent.
no slowing
slowing down
down or speeding
speeding
up . The
the same
up.
The total
total goes
goes up by the
same amount
amount each
each time
time..
Now
every month
instead of adding
The total
total
N
o w imagine
imagine that
that Ann
Ann takes
takes 5 dollars
dollars out
o u t of her
her piggybank
piggybank every
month instead
adding to it. The
balance
decrease by
a constant
decay. The
total amount
amount A in
balance would
would decrease
by a
constant amount
amount each
each month,
month, resulting
resulting in linear
linear decay.
The total
the piggybank
piggybank could
then be
the
could then
be modeled
modeled by
z 100 -‐ lOt
A =
The
The graph
graph of such
such an
an equation
equation is aa line
line with
with aa negative
negative slope.
slope.
A
Both exponential
linear decay
a negative
negative association
association between
between ttwo
things. As one
exponential decay
decay and
and linear
decay are
are instances
instances of a
w o things.
one
thing
thing decreases
absences over
over the
the semester
and final
thing increases,
increases, the other
other thing
decreases.. For example,
example, the number
number of absences
semester and
exam scores:
exam
Final Exam
Exam Score
Final
•
•••••
• • ••
•• • • •
Number of Absences
Number
Absences
When the
data points
points are
are close to forming
forming aa smooth
smooth line
line or graph
relationship, we
we can
can
When
the data
graph that
that shows
shows the negative
negative relationship,
that there
strong negative
negative association
association..
say that
there is aa strong
26
THE COLLEGE
THE
COLLEGE PANDA
PANDA
Both exponential
are instances
between two
t w o things.
things. As
As
exponential growth
growth and
and linear
linear growth
growth are
instances of a
a positive
positive association
association between
one
thing
increases,
the
other
thing
also
increases.
For
example,
the
number
of
hours
spent
studying
and
final
one thing increases, the other thing also increases .
example,
number
hours spent studying and
exam
scores::
exam scores
Final Exam
Exam Score
Final
•
•
••
•
•
•• • • •• •
Hours
Hours Studied
Studied
The graph
The
shows a positive
positive association
association that
that is quite
graph above
above shows
quite strong.
strong.
27
27
CHAPTER 3 EXPONENTIAL
GROWTH
CHAPTER
EXPONENTIAL V5.
VS. LINEAR
LINEAR GROWTH
CHAPTER
start on
CHAPTE R EXERCISE:
EXERCISE : Answers
Answers for this
this chapter
chapter start
on page
page 278.
A calculator should NOT be use d on the
followi ng questi ons.
The employees at a new bookstore must stock a
certain number of shelves so that the store is
ready for its
ready
w o weeks.
its opening
opening in ttwo
weeks. The
The .
employees
rate
employees stock
stock shelves
shelves at
at a
a constant
constant rate
throughout the
throughout
is the
the number
the two
two weeks.
weeks. If p(t) is
number
shelves left
of shelves
be stocked
stocked after
which of
left to be
after tt days,
days, which
the
statements best
best describes
the
the following
following statements
describes the
function p
function
p??
The value
The
value of aa house
the
house decreased
decreased by 8°/o
8% from
from the
previous year
previous
consecutive years.
Which of
year for n consecutive
years. Which
the following
the
value of
following graphs
graphs could
could model
model the
the value
the house
house over
over this
this time
time period?
period?
the
A)
A)
The function
A) The
function p is an
an increasing
increasing exponential
exponential
fw1ction.
function.
B) The
The function
function p
pisis a
a decreasing
decreasing exponential
exponential
hmction.
function.
C) The
The function
C
linear
function p is an
an increasing
increasing linear
function .
function.
Time
Time
D
The function
is aa decreasing
decreasing linear
D) The
function p is
linear
B)
B)
function.
function .
.-0
.-0
.-0
.-0
.-0
If the initial population of rats was 20 and grew
to 25 after the first year, which of the following
functions best models the population of rats P
with respect to the number of years t if the
population growth of rats is considered to be
exponential?
.-0
Time
C)
A) PP==5t
5t ++ 20
20
B)
B) P =
= 20(1.25)'
20(1.25)1
C)
: 20(5)'
C) P =
20(5) 1
D)
= 5:2
+ 20
D) P =
5t 2 +
Time
D)
D)
If the initial population of pandas was 100 and
grew to 125 after the first year, which of the
following functions best models the population
of pandas P with respect to the number of years
t if the population growth of pandas is
considered to be linear?
Time
A
5t+
00
A)) PP == 225t
+1100
B)
=100(1.25)'
B) P =
100(1.25) 1
C) P = 100(1.2)
100(1.2)'1
D) P=20t2+5t+100
D)
P = 20t 2 + 5t + 100
28
28
THE COLLEGE
PANDA
THE
COLLEGE PANDA
)I
flf(t)t ) =
:20(1+%)t
20 ( 1 + 100
15
N
= 1, 000(0.97) 41'
scientist uses
equation above
above to model
model the
the
A scientist
uses the
the equation
number of bacteria
Ninin aa petri
petri dish
dish after
after h
number
bacteria N
hours. According
According to the
the model,
model, the
the number
number of
hours.
bacteria is predicted
predicted to decrease
3% every
every kk
bacteria
decrease by 3°/o
minutes. What
What is the
the value
minutes.
value of kk??
The
models the
The function
function f above
above models
the temperature,
temperature, in
degrees
degrees Celsius,
Celsius, of a metal
metal alloy
alloy used
used in an
an
experiment,
experiment, where
where t is the
the number
number of seconds
seconds
after the
after
the experiment
experiment began
began.. Which
Which of the
the
following
the best interpretation
following is the
interpretation of the
the
number 15 in this
number
this context?
context?
1
1
A) A)
4
4
B) 4
A) The temperature,
temperature, in degree
degreess Celsius,
Celsius, of the
the
metal
beginning of the
metal alloy
alloy at
at the
the beginning
the
experiment
experiment
C)) 115
C
5
D) 240
B
The increa
increase
the temperature,
temperature, in degrees
degrees
B) The
se in the
Celsius,
the metal
Celsius, of the
metal alloy
alloy every
every 100
100
seconds during
during the
the experiment
experiment
seconds
C) The
The percent
which the
the temperature,
temperature, in
percent by which
degrees Celsius,
metal alloy
alloy
degrees
Celsius, of the metal
decreased
each second
next
decreased from
from each
second to the next
during
during the
the experiment
experiment
cC = 40.002)“
4(1.002) 21
The equation
above can
be used
used to model
model the
The
equation above
can be
the
number
cars, in millions
millions,, registered
number of cars,
registered in aa
certain
years after
According to the
certain state
state tt years
after 2009. According
the
model,
the
number
of
cars
registered
the state
state
model, the number
cars registered in the
n%every
every 6 months
months..
is projected
projected to increase
increase by n%
What is the
the value
value of nn ?
?
What
percent by which
which the temperature,
temperature, in
D) The percent
degrees
metal alloy
degrees Celsius,
Celsius, of the metal
alloy
increased
each second
second to the
the next
next
increased from each
during
experiment
during the experiment
A) 0.002
B) 0.04
C) 0.2
c u ) == 80(2)
80(2)é~
C(t)
D) 2
how aa certain
certain virus
virus spreads,
spreads,
To examine
examine how
scientists introduced
introduced the
virus to cells in aa test
test
scientists
the virus
tube
and found
the number
number of infected
infected cells
tube and
found that
that the
in the
the test tube
tube grew
exponentially over
grew exponentially
over time.
time.
The
function
The function C above
above models
models the
the number
number of
infected
infected cells in the
the test
test tube
tube t days
days after
after the
the
virus
on the
virus was
was introduced
introduced.. Based
Based on
the function,
function,
which of the
the following
which
following statements
statements is true?
true?
The
trees in aa forest
forest has
has been
The population
population of trees
been
decreasing
years.. The
decreasing by 6 percent
percent every
every 4 years
population
population at
at the
the beginning
beginning of 2015 was
was
estimated
14,000. If
If P represents
represents the
the
estimated to be
be 14,000.
years after
after 2015, which
which of
population of trees
trees tt years
population
the following
following equations
equations gives
the population
population of
the
gives the
trees
trees over
over time?
A) The predicted
predicted number
number of infected
infected cells in
test tube
the test
tube doubled
doubled every
every 5 days.
days.
B) The predicted
predicted number
number of infected
infected cells in
the test tube
tube grew
grew by aa factor
factor of 5 every
every two
two
days.
da ys.
infected cells in
C) The
The predicted
predicted number
number of infected
the test tube
tube doubled
doubled every
every day.
D) The predicted
D
predicted number
number of infected
infected cells in
the test tube
tube grew
the
grew by aa factor of 5 every
every day.
day.
I
A) P
P = 14,000
14,000(0.06)t
A)
(0.06):J
B) P =
= 14,000
14,000 +
+ 0.94(4t)
B)
0.94(4t)
C) P
= 14,000 (0.94) 41
C)
P=14,000(0.94)4'
I
= 14,000
(0.94)4°
D) P
P=
14,000(0.94)é
29
29
CHAPTER 3 EXPONENTIAL VS. LINEAR GROWTH
A calculator is allowed on the following
questions.
Jamie
some money
Jamie owes
owes Tina some
money and
and decides
decides to pay
pay
her back
Tina receives
receives 33
her
back in the
the following
following way.
way. Tma
dollars
dollars the
the first
first day,
day, 66 dollars
dollars the
the second
second day,
day, 18
18
dollars
day, and
54 dollars
dollars the third
third day,
and 54
dollars the
the fourth
fourth
day. Which
Which of the
the following
describes the
the
day.
following best
best describes
relationship
relationship between
between time
time and
and the
the total
total amount
amount
of money
(cumulative) Tina
from
money (cumulative)
Tma has
has received
received from
Jamie
over the
the course
course of these
these four
Jamie over
four days?
days?
Which scatterplot
the strongest
Which
scatterplot shows
shows the
strongest positive
positive
association
association between
between x and
and y?
y?
A)
y
.,.
•••
.:..
.
•,,
••••
.
,, ..
......
••
• ••
•
A)
B)
C)
D)
Increasing linear
linear
Increasing
Decreasing linear
linear
Decreasing
Exponential growth
Exponential
growth
Exponential decay
Exponentialdecay
X
B)
y
.. -••
.-··.
......,.
•••
•
•
Albert has a large book collection . He decides to
trade in two of his used books for one new book
each month at a local bookstore. Which of the
following best describes the relationship
between time (in months) and the total number
of books in Albert's collection?
•
..,..
• • • ••
•• •
•
A)
B)
C)
D)
X
C)
y
,-
.,.,,,.
·"
~·
.J:
Increasing
Increasing linear
linear
Decreasing
Decreasing linear
linear
Exponential
growth
Exponential growth
Exponential
decay
Exponential decay
A scientist
scientist counts
counts 80 cells in aa petri
petri dish
dish and
and
finds
into two
two new
n e w cells
finds that
that each
each one
one splits
splits into
cells
every
uses the
= cr’
every hour.
hour. He
He uses
the function
function A(
A ( t) =
cr1 to
calculate
calculate the
the total
total number
number of cells
cells in the
the petri
petri
dish
hours. Which
Which of the
the following
following
dish after
after t hours.
assigns
values to c and
and r ??
assigns the
the correct
correct values
X
D)
y
A) c =
= 40,
= 2
40,rr =
B)
c
=
80,r
=
8)
80, r = 0.5
,~ ....
,, J.
C) cC =
: 11.5
.5
= 80,r
80, r =
D) cc = 80,
= 2
80,rr =
·-"~'y.
•• ti'•
•
•
X
30
THE
PANDA
COLLEGE PANDA
THE COLLEGE
ID,____
_____
.__....____,_
.....
The
dollars, of
price P, in dollars,
the price
shows the
below shows
table below
The table
a
beginning of
after the beginning
days after
cntde oil t days
of crude
barrel of
a barrel
an
shortage.
an oil shortage.
Of the following scenarios, which one would
result in linear growth of the square footage of a
store?
A)
footage by
square footage
increases the square
owner increases
The owner
A) The
year.
each year.
% each
0.75
0.75%
footage by
square footage
increases the square
owner increases
8) The owner
B)
each year.
5% each
5%
of the
store by 5%
the store
expands the
owner expands
The owner
C) The
5%of
the
each year.
footage each
original square
original
square footage
year.
between adding
alternates between
owner alternates
The owner
D) The
adding 200
the
square feet the
and 300 square
year and
one year
square feet one
square
year.
next year.
next
Price,
(dollars)
Price, P (dollars)
0
50.00
15
60.51
30
73.22
45
88.77
11 11
the
model the
used to model
m(2) " is used
equation P = m(2)'/
the equation
If the
which of the
and P, which
between t and
relationship between
relationship
?
and n ?
values of m and
the values
be the
could be
following could
following
11 = 54.38
and n
m = 25 and
A) m
86.12
25and n =
B) m = 25andn
=86.12
50 and n = 54.38
C) m = 50andn
C)
and n =
= 50 and
m =
D) m
= 86.12
equipment
an equipment
operation, an
year of operation,
its first year
During its
During
line .
product line.
items in its
carried 6,400 items
supplier
supplier carried
its product
carried
the supplier
next 66 years,
each of the next
For each
years, the
supplier carried
number of items
the number
half the
product line half
in its product
items it
type of
What type
previous year. What
the previous
carried the
had carried
had
the
items the
number of items
the number
model the
best to model
model is best
model
given
any given
line for any
carried in its product
supplier
supplier carried
product line
operation?
years of operation?
year
year in its first 7 years
A)
B)
B)
C)
D)
Number
days, t
Number of days,
a
stored in a
gigabytes, stored
data, in gigabytes,
amount of data,
The amount
If 16
hours. If
every 15 hours.
database
increases by 2% every
database increases
stored in the
gigabytes
the
currently stored
data is currently
worth of data
gigabytes worth
functions g
database,
following functions
the following
which of the
database, which
will
that will
gigabytes, that
gives
data, in gigabytes,
amount of data,
the amount
gives the
now?
be
days from now?
database t days
the database
stored in the
be stored
model
An exponential
growth model
exponential growth
An
model
decay model
exponential decay
An exponential
A linear
model
growth model
linear growth
A linear
model
decay model
linear decay
16(2)1s1
g(t) = 16(2)15'
A) 80)
A)
I
= 16(1.02)15
B) g(t)
B)
g<t>=16(1.02)f’-<
51
C)
g(t)= 16(1.02)s
C) g(t)=16(1.02)%
V
= 200(i
V
= 1,500t
81
D)
g(t) = 16(1.02) '°
D) g(t)=16(1.02)%
)
An analyst
wo
the ttwo
accurate the
how accurate
evaluating how
analyst is evaluating
An
number
models
total number
the total
predicting the
are in predicting
above are
models above
of views,
days after
after
receives t days
video receives
online video
an online
V, an
views, V,
are
views are
more views
itit is released.
many more
How many
released. How
predicted
the
than by the
model than
linear model
the linear
predicted by the
exponential
video is
the video
after the
days after
model 4 days
exponential model
released?
released?
A) 1,400
B) 2,800
C) 3,200
D) 4,000
I
31
31
- ----
Rates
Rates
I’ve found
rate problems
problems to
students just
just “get"
intuitively, others
completely
I've
found rate
to be
be pretty
pretty polarizing‐some
polarizing-some students
"get" them
them intuitively,
others get
get completely
Most of the
rate problems
be pretty
straightforward, but
aren’t, II highly
highly
lost. Most
the rate
problems on
on the SAT will be
pretty straightforward,
but for the ones
ones that
that aren't,
recommend using
up the
the solution
solution ((if
i f you’ve
you should
should
recommend
using conversion
conversion factors to set up
you've gone
gone through
through chemistry,
chemistry, you
know
I ' m talking
factors are
approach a lot of these
these problems,
problems,
know what
what I'm
talking about).
about) . Conversion
Conversion factors
are a fool-proof
fool-proof way
way to approach
but
be slow-going
solvers.. I’ll
both the
the straightforward,
straightforward, intuitive
intuitive
but they can
can be
slow-going for stronger
stronger problem
problem solvers
I'll be
be covering
covering both
approaches
approach throughout
examples in this
this chapter
chapter..
approaches and
and the
the conversion
conversion factor approach
throughout the examples
EXAMPLE 1: A bicycle
bicycle manufacturer
manufacturer can
can produce
produce 20
20 bicycles
bicycles per
per hour
hour.. How
H o w many
many hours
hours would
would it
it take
take
EXAMPLE
the
manufacturer to produce
produce 320
320 bicycles?
bicycles?
the manufacturer
Wedivide
divide the
the total
+ 20
20 =
: [!§]hours
hours..
Easy enough.
enough . We
total by the rate
rate to get 320 ...,...
EXAMPLE 2:
2: A rocket
rocket has
has 360
360 gallons
fuel left
left after
after 2 hours
hours of flight,
flight, and
and only
only 100
100 gallons
gallons after
hours
EXAMPLE
gallons of fuel
after 66 hours
of filght.
flight. It burns
burns n71gallons
gallons of fuel
every hour
hour of flight,
flight, where
where n is a constant.
constant. What
What is the
value of n ??
fuel for every
the value
Here,
we are
are figuring
figuring out
o u t the
the rate.
rate. ln
In 6 -‐ 2 =z 4 hours
hours of flight, the rocket
Here, we
rocket burned
burned 360 -‐ 100 =z 260 gallons
gallons of
260 ILt:l
Therefore, the
the rocket
rocket bums
z ~
5] gallons
gallons of fuel every
fuel. Therefore,
bums a? =
every hour.
hour .
4
EXAMPLE 3: A box
at the
the supermarket
can hold
hold 66 oranges
oranges each.
Each orange
20 cents
cents.. Given
Given that
that
EXAMPLE
box at
supermarket can
each. Each
orange costs
costs 20
the
has a
budget of $540
to stock
oranges, how
boxes will
will the
the supermarket
supermarket be
be able
able to
to
the supermarket
supermarket has
a budget
$540 to
stock oranges,
how many
many boxes
fill?
If each
dollar would
would be
be enough
enough for 5 oranges.
Five hundred
forty dollars
would then
If
each orange
orange is 20 cents,
cents , then
then aa dollar
oranges . Five
hundred forty
dollars would
then
be
enough for 540 x 5 =z 2700 oranges,
oranges, which
which would
would fill 2700 ...,...
+ 6=
: j 450 Iboxes.
be enough
boxes.
above were
were quite
straightforward and
and didn
didn’t
out conversion
factors, but
but
The examples
examples above
quite straightforward
't really
really call for writing
writing out
conversion factors,
what if
if we wanted
conversion factors
factors for Example
Example 3?
3? What
what
wanted to use
use conversion
What would’ve
would 've the
the solution
solution looked
looked like?
IOOcents
l g~
ange
ox
.centS" 1
1l bbox
5 4.de-Ha"fs
0 W xx 100.deHar
‐ 4450
0 boxes
540
x
x ---=
l1d o l i a r? X 20
2 .centS"
0 m X 66ganges
5 boxes
~
32
THE
THE COLLEGE
COLLEGE PANDA
PANDA
As
help you
go from
from one
one set
set of
to another
As you
you can
can see,
see, conversion
conversion factors
factors are
are multipliers
multipliers that
that help
you go
of units
units to
another.. They're
They’re
1
.
.
.
.
.
.
oran e
usually
as fractions,
the question
question (e.g.
usually expressed
expressed as
fractions, and
and they
they represent
represent either
either information
information provided
prov1ded by the
(e.g. orange)
20 cents
,
100 cents)
cents Wh
..
.. f
bl em, you
._
((e.g.
100
1ve a pro
d ar d unit
or stan
standard
u n .i t conversmns
) .. When
usmg
c o n v e r s 1 o n factors
solve
problem,
you must
must set
set
conversions
e.g. m
dollar
en using
conversion
actors to so
1
them
and denominators.
them up
up in the
the right
right sequence
sequence and
and with
with the
the appropriate
appropriate numerators
numerators and
denominators.
Wags)
The
teach you
they're used,
even
The rest
rest of the
the examples
examples in this
this chapter
chapter are
are done
done with
with conversion
conversion factors
factors to teach
you how
how they’re
used, even
though
to solve
the problems
though you
you may
may be
be able
able to
solve the
problems "intuitively".
“intuitively”.
EXAMPLE
travel 1 mile
second-s. At
Afthis
many miles
can the
ear
EXAMPLE 4: A car
car can
can travel
mile in 1 minute
minute and
and 15
15seconds.
this rate,
rate, how
how many
miles can
the car
travel
travel in 1 hour?
hour?
In most
asking for. We
convert that
hour to
most rate
rate problems,
problems, you'll
you’ll start
start with
with what
what the
the question
question is asking
We need
need to convert
that 11 hour
to aa
distance
distance that
that the
the car
car travels
travels.. The
The car's
car’s rate
rate is 1 mile
mile every
every 75 seconds
seconds..
1
r x 6 0 m m 60seeonfis
x 60
miles =
:
11 mile
mile
= 60
60 x
60 miles
I miles
mil
1 hoof x 60 .mifl1:rtes x 60 _seetmcisx
148
r
1wear
hoof
1.rnimrte
75 _seetmcis
75
es
xlmimifixfiseeerfisz
75
The
are canceling,
are you're
doing things
The units
units should
should cancel
cancel as
as you
you go
go along.
along. If the
the units
units are
canceling, chances
chances are
you're doing
things right.
right.
Notice
up with.
with . This
This is
another sign
that we’ve
we've
Notice that
that the
the "miles"
”miles” unit
unit at
at the
the end
end is
is the
the unit
u n i t we
we wanted
wanted to end
end up
is another
sign that
done
done things
things correctly.
correctly.
EXAMPLE 5: Tom
Tom drives
drives 30
30miles
miles at
at an
an average
average rate
rate of 50
50 miles
miles per
per hour.
hour. If
If Leona
Leona drives
drives at
EXAMPLE
at an
an average
average
rate
same distance?
distance?
rate of 40
40 miles
miles per
per hour,
hour, how
how many
many more
more minutes
minutes will
w i l l it take
take her
her to travel
travel the
the same
We have
have to figure
figure out
o u t how
how long
takes Tom to
to drive
drive 30
miles::
We
long it takes
30 miles
hour
60 minutes
minutes
.
1 hour
.
30 miles
36 minutes
m11es><
x
.
x
h
=
miles
1 hour
2 36 minutes
1es x
our
50 m1
Leona
will take
take
Leona will
x
30 miles
miles><
hour
~1 ho_
t
x
miles
4 0 m1
es
60 minutes
60
i°~inutes = 45 minutes
1 hour
: 45 minutes
our
So,
So,
45-‐ 36
=~
E minutes
36 =
minutes
45
EXAMPLE 6: To prepare
prepare for class,
class, Mr.
Mr. Chu
Chu has
has to print
print a
number of booklets
booklets with
with p pages
pages per
per booklet.
booklet. If
If
EXAMPLE
a number
every 5 pages
pages cost
cost c0 cents
cents to
to print
print and
and he
he spent
spent aa total
total of d dollars,
dollars, how
booklets did
print
every
how many booklets
did Mr. Chu
Chu print
p,c,
and d ??
in terms of p,
c, and
cp
A)
A) 500d
500d
100d
B) 100d
500d
C) 500d
B) cp
C) cp
,.J _
d
D) cp
pages x 11booklet
SOOd b
booklets
55 ~
booklet I 500d
kl
X
=
-00
ets
r s x lIda-Harxccents
ppages
cp
.1xrue1r
c.centS'
p~
cp
,i---:::-
d ~ctrS
D) 35
Sd
X
1OOcents‘
100
.cetrtS'
..J-J.J...-::°
X
CT§]
.
Answer (C) .
Answer
33
33
CHAPTER 4 RATES
CHAPTER
RATES
CHAPTEREXERCISEAnswers
:
for this chapter star t on page 280.
A calculator should
O T be
should N
NOT
be used
used on
on the
the
following questions.
following
questions.
submarine descends
As a submarine
descends into
into the
the deep
deep ocean,
ocean,
the pressure
pressure itit m
the
u s t withstand
increases.
must
withstand increases. At
At an
an
altitude of ‐700
altitude
- 700 meters,
meters, the
the pressure
pressure is
is 50
50 atrn
atrn
(atmosphe res), and
(atmospheres),
and at an
an altitude
altitude of
of -‐900
900
meters, the
the pressure
meters,
For every
pressure is 70 atrn.
atrn . For
every 10
10
meters the
meters
the submarine
submarine descends,
descends, the
the pressure
pressure itit
faces increases
increases by n, where
where n is
is a
a constant.
constant. What
What
is the value
va lue of n ?
?
Tim's diet
diet plan
plan calls for 60
Tim’s
60 grams
grams of
of protein
protein per
per
day.
If
Tim
day. If
were
were to meet
meet this
this requirement
requirement by
by
only eating
eating a certain
only
certain protein
protein bar
bar that
that contains
contains 30
30
grams of protein,
grams
protein, how
how many
many protein
protein bars
bars would
would
he have
have to buy
he
buy to last a
a week?
week?
A) 0.1
B) 1
C) 2
D
0
D)) 110
An empty
empty pool
pool can
can be
hours if
be filled in 5 hours
if water
water is
is
pumped in at 300 gallons
pumped
gallons an
an hour.
hour . How
How many
many
hours would
would it take
take to fill the
hours
the pool
pool if water
water is
is
pumped in at 500 gallons
pumped
ga llon s an
an hour?
hour?
An electronics
electronics company
company sells
sells computer
computer monitors
monitors
and
releases a
and releases
e w model
a nnew
model every
every year.
year . With
With each
each
new model,
model, the
the company
company increases
new
increases the
the screen
screen
size by a constant
constan t amount.
size
amount. In
ln 2005, the
the screen
screen
size was
was 15.5 inches.
inches . In
size
was
1n2011, the screen
screen size
size was
inches. Which
Which of the
18.5 inches.
the following
following best
best
describes how
describes
how the
the screen
screen size
size changed
changed between
between
2005 and
and 2011?
A) The
The company
company increases
A)
size by
increases the
the screen
screen size
by
0.5 inch
inch every
every year.
year .
The company
company increases
B) The
size by 1
increases the
the screen
screen size
inches every
inches
every year.
year.
company increases
C) The company
increases the
the screen
screen size
size by
by 22
inches
inches every
every year.
year .
The company
D) The
company increases
increases the
the screen
screen size
size by
by 3
3
inches
inches every
every year.
If a app les cost d dollars, which of the following
expressions gives the cost of 20 apples, in
dollars?
20a
A) 303
d
d
B)
B)
y20d
a
a
C)
C) 207
20d
20
D) ad
D>H
34
THE COLLEGE
THE
COLLEGE PANDA
PANDA
During
During a
a raceon
race on a
a circular
circular race
race track,
track, aa racecar
racecar
Idina
can type
90 words
words in 2.5 minutes.
ow
Idina can
type 90
minutes. H
How
many words
type in 12
many
words can
can she
she type
12 minutes?
minutes?
‘
bums fuel
fuel at
at a
a constant
constant rate.
bums
rate. After
After lap
lap 4, the
the
racecar has
tank. After
After lap
racecar
has 22 gallons
ga110ns left
left in its
its tank.
lap 7,
the racecar
gallons left
the
racecar has
has 18 gallons
left in its
its tank.
tank.
Assuming
the racecar
does not
not refuel,
refuel, after
after
Assuming the
racecar does
which lap
will the
racecar have
have 6 gallons
gallons left
which
lap will
the racecar
left in
tank?
its tank?
A) Lap
13
Lap 13
B) Lap
B)
Lap 15
15
C) Lap
Lap 16
16
C)
_
D) L
D
Lap 19
19
) ap
salesman at aa tea company
company makes
makes a
a $15
A salesman
commission on
on every
every $100 worth
worth of products
products
commission
that he
he sells.
sells. If
If a
a jar of tea leaves
leaves is $20, how
how
that
many jars
jars would
would he
he have
have to
to sell
sell to
to make
many
make $180 in
commission?
commission?
_
By 1:00 PM,
PM, a total
total of 40
boxes had
had been
40 boxes
been
unloaded from
a delivery
delivery truck.
truck . By 3:30 PM,
a
unloaded
from a
PM, a
total of 65
65 boxes
boxes had
had been
been unloaded
unloaded from
from the
the
total
same truck.
truck. If
If boxes
boxes are
are unloaded
unloaded from
from the
same
truck
the truck
constant rate,
at a constant
number of
rate, what
what is the
the total
total number
boxes that
that will
have been
been unloaded
unloaded from
boxes
will have
from the
the
truck by
by 7:00 PM?
truck
PM?
A train
it
train covers
covers 32
32 kilometers
kilometers in 14.5 minutes.
minutes . If
If it
continues
at the
the same
rate, which
which of the
continues to travel
travel at
same rate,
the
following
closest to
the distance
distance it
will travel
travel
following is
is closest
to the
it will
in 2 hours?
hours?
54kilometers
A) 54
kilometers
A calculator is allowed
following
allowed on the following
questions.
questions.
B) 265 kilometers
kilometers
C)
364
kilometers
C) 364 kilometers
D) 928 kilometers
kilometers
A rolling
covers a
feet in
in 4
rolling ball
ball covers
a distance
distance of 2400 feet
minutes.
speed, in
minutes . What
What is the
the ball’s
ball 's average
average speed,
inches
er second?
12 inches
inches per
second? (12
inche s =
= 1 foot
foot)
P
(
)
I1
_
Peanut
an industrial
container at
Peanut oil
oil leaks
leaks out
out of
of an
industrial container
at
the
If the
the rate
rate of 3 liters
liters in 2 hours.
hours. If
the peanut
peanut oil
oil
costs
costs 8 dollars
dollars per
per liter,
liter, how
how many
many dollars’
dollars' worth
worth
will
will be
be lost
lost in 11 hours?
hours?
A)
A) $60
B)
B) $96
C) $118
$118
D)
D) $132
35
CHAPTER
CHAPTER 4 RATES
RATES
A recipe for soap calls for 1~ cups of lye for
Henry
30 miles
per hour
hour and
Henry drives
drives 150 miles
miles at
at 30
miles per
and
then
miles per
per hour.
then another
another 200 miles
miles at
at 50
50 miles
hour .
What
his average
speed, in miles
What was
was his
average speed,
miles per
per hour,
hour,
for the
journey, to the
the entire
entire journey,
the nearest
nearest hundredth?
hundredth?
every 2 cup
cup of castor
castor oil. How
H o w many
lye
every~
many cups
cups of lye
are needed
batch of soap
soap that
cups of
are
needed for aa batch
that uses
uses 3 cups
castor
oil?
castor
4
A) 38.89
B) 40.00
5
C) 42.33
A)
1
A) 15
B) 5
D) 43.58
9!4
C)
C) 9g
5
1
D)
D) 11~
113
4
A ”slow”
"slow" clock falls behind
behind at
at the
the same
same rate
rate
It is set
set to the
the correct
correct time
time at 4:00
every hour. It
every
A M . When
When the
AM.
the clock shows
shows 5:00 AM the
the same
same
AM . When
When the
day, the
correct time
day,
the correct
time is 5:08 AM.
the clock
shows 10:30 AM
AM that
day, what
what is the
the correct
correct
shows
that day,
An 8 inch by 10 inch piece of cardboard costs
$2.00. If the cost of a piece of cardboard is
proportional to its area, what is the cost of a
piece of cardboard that is 16 inches by 20 inches?
time?
time?
A) 11:02 AM
AM
AM
B) 11:18 AM
AM
C) 11:22 AM
D) 12:18 PM
D)
A) $4.00
B) $8.00
C) $12.00
D) $16.00
Jared
are tasked
Jared and
and Robert
Robert are
are accountants
accountants who
who are
tasked
with
takes Jared
Jared
with reviewing
reviewing financial
financial reports.
reports. It takes
15 hours,
rate, to review
15
hours, working
working at a constant
constant rate,
review a
= 2 large bahar
400 kulack = 29 pikol
9 pikol
report
financial
pages of financial
report containing
containing 240 pages
statements.
works at twice Jared
Jared’s
statements. If
If Robert
Robert works
's rate,
rate,
how
many minutes
minutes would
would it take
Robert to
how many
take Robert
review
pages of financial
financial
review a
a report
report containing
containing 120 pages
statements?
statements?
The formulas
above represent
represent the
relationships
The
formulas above
the relationships
between some
some units
weight that
were once
once
between
unit s of weight
that were
used in Indonesia
Indonesia.. A weight
weight of 1,000 kulack
used
kulack is
equivalent
how many
many large
large bahar?
(Round
equivalent to how
bahar? (Round
your answer
answer to the
the nearest
nearest whole
whole number.)
number.)
your
A) 225
B) 345
C) 450
D) 900
36
36
COLLEGE PANDA
THE COLLEGE
PANDA
A flask contains an acidic solution with a
concentration of 7.1 x 1015 hydrogen ions per
mi!Liliter. If 4.8 x 1023 hydrogen ions have a total
mass of 0.8 grams, which of the following is
closest to the concentration, in grams per liter, of
the acidic solution?
5
A) 1.2 ><10_5
X 10A)
8
10B) 1.2 x
><1o‘8
-5
x 10
C) 1.5 ><10_5
8
D) 1.5 ><10_8
X 10D)
Brett currently spends $160 each month on gas .
His current car is able to travel 30 miles per
gallon of gas. He decides to switch his current
car for a new car that is able to travel 40 miles
per gallon of gas. Assuming the price of gas
stays the same, how much will he spend on gas
each month with the new car?
A) $100
B)
8) $120
$120
C) $130
D)
D) $140
Margaret can buy 4 jars of honey for 9 dollars ,
and she can sell 3 jars of honey for 15 dollars.
How many jars of honey would she have to buy
and then sell to make a total profit of 132
dollars?
37
Ratio & Prop
Ratio
Proportion
ortion
Let’s
parking lot
lot is
is 5:2
5:2 (5
Because ratios
ratios can
be written
as fractions,
fractions,
Let's say that
that the
the ratio
ratio of cars
cars to trucks
trucks in
in aa parking
(5 to
to 2).
2). Because
can be
written as
this
a t i o is
15equrvalent
to ~.
2 A
A ratio
r a t i o of
of 5:2
5:2 is
15also
a t i o of
to the
the
this rratio
equivalent to
also equrvalent
equivalent to
to a
a rratio
of 10:4,
10:4, since
since the
the latter
latter reduces
reduces to
former.
former.
In
ratio of 5:2 means
means that
every 5 cars,
And assuming
Ln this
this context,
context, a ratio
that for every
cars, there
there are
are 22 trucks.
trucks . And
assuming that
that there
there are
are only
only
cars and
ratio also
means that
that there
vehicles. By
By the
same
cars
and trucks
trucks parked
parked in the
the lot,
lot, the
the ratio
also means
there are
are 5
5 cars
cars for
for every
every 77 vehicles.
the same
token,
vehicles.
token, there
there are
are 2 trucks
trucks for every
every 7 vehicles.
EXAMPLE1: Minyoung
EXAMPLE
Minyoung bought
bagels for
event. The
The ratio
ratio of
the number
number of
of
bought croissants
croissants and
and bagels
for a
a breakfast
breakfast event.
of the
croissants
bought to the
number of
of bagels
bagels she
she bought
bought was
[f Minyoung
Minyoung bought
bought 72
bagels, how
how
croissants she
she bought
the number
was 33 to
to 4.
4. If
72 bagels,
many croissants
did she
she buy?
buy?
many
croissants did
According to the
Minyoung bought
bought 3 croissants
croissants for
bought 72
According
the given
given ratio,
ratio, Minyoung
for every
every 4
4 bagels.
bagels. Since
Since she
she bought
72 bagels,
bagels, she
she
must
have bought
bought
must have
3 croissants
croissants
~
.
72 bagels
bagels =
: ~
54 croissants
x 72
croissants
bagels
4 bage
1s
EXAMPLE
EXAMPLE 2: Arfand
Arfand is
is following
following a
recipe for
for a
seasoning blend
blend that
black pepper,
pepper, and
and
a recipe
a seasoning
that requires
requires sea
sea salt,
salt, black
paprika.
paprika. According
According to the
the recipe,
recipe, the
the ratio
ratio of
of grams
grams of
of sea
sea salt
pepper should
1:2,
and
salt to
to grams
grams of
of black
black pepper
should be
be 1:2, and
ratio of grams
grams of black
black pepper
pepper to
of paprika
paprika should
should be
How many
paprika should
should
the
the ratio
to grams
grams of
be 4:3.
4:3. How
many grams
grams of
of paprika
Arfand
use to make
make 108 grams
seasoning blend?
blend?
Arfand use
grams of
of the
the seasoning
Since black
black pepper
both the
”common basis
basis”" for
for
Since
pepper is involved
involved in both
the given
given ratios,
ratios, we
we can
can use
use itit to
to establish
establish aa "common
comparison. First,
comparison.
black pepper
pepper ratio
ratio by
get
2:4.
Why
multiply
by
2?
Because
First, multiply
multiply the
the sea salt
salt to black
by 2
2 to
to get 2:4. Why multiply by 2? Because
now
n
o w the
“ 4 " in the
lines up
with the
” 4 ” in
in the
the black
black pepper
pepper to
the "4"
the ratio
ratio lines
up with
the "4"
to paprika
paprika ratio.
ratio. Once
Once they
they are
are lined
lined up,
up , we
we
can establish
can
that the
between the
three
ingredients
is
2:4:3
(sea
salt
to
black
pepper
to
paprika).
establish that
the ratio
ratio between
the three ingredients is 2:4:3 (sea salt to black pepper to paprika).
According
be used
used for
for every
+4
+ 33 =
: 99 grams
of the
blend.. Therefore,
Therefore,
According to this
this ratio,
ratio , 3 grams
grams of
of paprika
paprika should
should be
every 2
2+
4+
grams of
the blend
3
Arfand should
should use
use ~
5 xx 108 =
: 36 grams
grams of
paprika.
Arfand
of paprika.
lliJ
38
38
COLLEGE PANDA
THE COLLEGE
PANDA
Proportion
Proportion
addition to ratios,
the SAT will
In addition
ratios, the
test you
you on
on proportions,
o t in the
that you
you typically
typically learn
will also
also test
proportions, but
but n
not
the way
way that
learn
them in school
school (direct
(direct vs.
vs. indirect
indirect proportion).
them
will give
and ask
you
how
proportion). Instead,
Instead, the
the SAT will
give you
you aa relationship
relationship and
ask you how
a change
change in one
one variable
a
variable affects another.
another.
Let's run
through a
a quick
example . Imagine
Let’s
r u n through
quick example.
Imagine we
we have
have a
a triangle.
triangle. We
We know
know that
that the
the area
area of
of a
a triangle
triangle is
is
A = ébh
Now
Now let's
let’s say
say we
we triple
triple the
the height.
height. What
What happens
happens to the
the area?
area?
Well, if we
we triple
triple the height,
new height
is 3h. The
The new
height, the new
height is
new area
area is
is then
then
1
Anew = 517611) 2 3
1
(ibh)
2 314.0“)
See what
what happened?
happened? The
rearranged so
so that
that we
e w area
is three
three times
times the
The terms
terms were
were rearranged
we could
could clearly
clearly see the
the nnew
area is
the
area. We
old area.
We put
put the ”3”
old formula.
"3" out
out in front
front of the
the old
formula.
This technique
This
extremely important
important because
because it saves
saves us
on tough
problems. We
We could’ve
technique is extremely
us time
time on
tough proportion
proportion problems.
could've
made
made up
up numbers
numbers for the
the base
base and
and the
the height
height and
and calculated
while
that’s
certame
calculated everything
everything out,
out, and
and while that's certainly a
a
strategy
strategy you
you should
should have
would’ve taken
longer
and
silly
have in your
your toolbox,
toolbox, it would've
taken much
much longer and left us
us more
more open
open to
to silly
mistakes .
mistakes.
Let's do
Let’s
do a
a few more
more examples.
examples.
EXAMPLE
The radius
radius of a
circle is increased
increased by
by 25°/o.
By what
EXAMPLE 3: The
a circle
25%. By
what percent
percent does
does the
the area
area of
of the
the circle
circle
increase?
mcrease?
Let the
be AAM.
the original
original radius
radius is
is r,
r, then
e w radius
radius is
is 1.25r.
the original
original area
area be
old· If
lf the
then the
the nnew
l.25r .
Am,
= 7T(1.25r)2 z (l.25)2(7rr2) =1.5625(7rr2) = 1562514,”
We can
56.25% I..
can see that
that the area
area increases
increases by j 56.25%
The idea
idea is to get a number
in front
of the old
old formula.
formula. In
turned out
1.5625.
The
number in
front of
In this
this example,
example, that
that number
number turned
out to
to be
be 1.5625.
Also note
1.25r was
was wrapped
in parentheses
parentheses so
Also
note that
that the
the l.25r
wrapped in
so that
that the
the whole
whole thing
thing gets
gets squared.
squared. It
It would’ve
would've been
been
incorrect
: rr(l.25
71'(1.25)r2
we wouldn't
wouldn’t be
e w radius.
incorrect to have
have Am”
A ,ww =
)r 2 because
because we
be squaring
squaring the
the nnew
radius.
EXAMPLE
4: The
The length
length of
is mcreased
increased by
by 20%.
width is
is decreased
EXAMPLE 4:
of aarectangle
rectangle is
20%. The
The width
decreased by
by 20%.
20%. Which
Which of
of the
the
following
accurately
describes
the
change
in
the
area
of
the
rectangle?
following accurately describes the change m the area of the rectangle?
A)
A) Increases
Increases by 10%
B) Decreases
Decreases by 10%
10%
C) Decreases
Decreases by 4%
4%
D) Stays
Stays the
the same
same
D)
Originally,
A=
: lw.
120. Now,
Originally, A
Now,
Anpzu
A, ww
Z (1.201)(
(1201)(0.80u’)
=
0.80w ) :=
§J .
0.96171)
0 .96/w
= 0.96A01d
0.96A 0 1d
The area
Answer (C) . Most
Most students
students think
(D). It's
It’s not.
The
area has
has decreased
decreased by 4%. Answer
think the
the answer
answer is
is (D).
not.
39
CHAPTER
CHAPTER 5 RATIO
RATIO & PROPORTION
PROPORTION
EXAMPLES:
EXAMPLE 5:
F_
= 9q1q2
94142
-
,2
7‘2
attraction between
between two
two particles
particles can
can be
be determined
determined by the
formula above,
The force of attraction
the formula
above, in which F is the
force between
q1 and
and 472
q2are
are the
charges of the
the two
between them,
them, rr is the
the distance
distance between
between them,
them, and
and q,
the charges
two particles.
particles. If
resulting force of attraction
the
the distance
distance between
between ttwo
w o charged
charged particles
particles is doubled,
doubled, the
the resulting
attraction is what
what fraction
fraction of
the
original force?
the original
1
Ali
l
3);
1
C’s
1
D)1‐6
F 2 94qu z<1>2(9q1qz) _1 (9q1qz)=1p
""“’
(2r)2
2
r2
4
r2
4 0“
Answer
how we do nnot
the formula
formula affect the
the result.
result. In
ln getting
Answer ~(B) . Notice
Notice how
o t let constants
constants like
like the "9"
" 9 ” in the
getting
aa number
number out
front, students
mixing that
that number
number up with
that were
out front,
students often
often make
make the
the mistake
mistake of mixing
with numbers
numbers that
were
originally in the
originally
the formula.
formula.
EXAMPLE
cube is tripled.
each side
side must
been increased
EXAMPLE 6: The
The volume
volume of aa cube
tripled. The
The length
length of each
must have
have been
increased by
approximately
approximately what
what percent?
percent?
A)3%
A) 3%
B}12%
B) 12°/o
C)33%
C)
33°/o
0)44%
D) 44%
Now
volume of a
a cube
s3 where
N o w we
we have
have to solve
solve backwards
backwards.. Keep
Keep in mind
mind that
that the
the volume
cube is V = S3
where ss is the
the length
length of
each
can still apply
apply the
process as
before: increase
each side.
side. Even
Even though
though this
this problem
problem is a little
little different,
different, we can
the same
same process
as before:
increase
each
side
by
some
factor
and
rearrange
the
terms
to
extract
a
number
.
Only
this
time,
we
have
to
use
x.
each side
some factor and rearrange the terms extract number. Only this time,
have use
3
= (xs)
(XS)
Vnew
Vm'w :
Vm’u' 2X
3 3 _
5 ‐X
3
Void
3
3 must be equal
. That
Notice
That xx3
new
Notice how
how we
we were
were still
still able
able to extract
extract something
something out
out in front,
front, xx3.
must be equal to 3 if
if the
the n
ew
volume
triple the
old volume
volume is to be
be triple
the old
volume::
3
xx3z3
=3
x:\3/§z1.44
= ~ ~ 1.44
X
Each
have been
Answer ~(D) .Each side
side must
must have
been increased
increased by
by approximately
approximately 44%. Answer
40
THE COLLEGE
THE
COLLEGE PANDA
PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 282.
A calculator should NOT be used on the
following questions .
A calculator is allowed
following
allowed on the following
questions.
questions.
The ratio
and the
the ratio
ratio of a to b is 7:6, and
ratio of b to c is
8:5. If
what is the
value of c ??
If a = 28, what
the value
P =-
v2
= RF
Electric
Electric power
voltage V and
power P is related
related to the
the voltage
and
resistance R by the
resistance
the formula
the voltage
voltage
formula above.
above. If
If the
were halved,
how would
would the
the electric
electric power
were
halved, how
power be
be
affected?
affected?
A) The electric
electric power
power would
would be
be 4 times
times greater.
greater .
A)
8) The electric
B)
greater..
electric power
power would
would be
be 2 times
times greater
C) The
The electric
electric power
power would
would be
be halved.
halved.
D) The
power would
would be
quarter of
The electric
electric power
be aa quarter
what
it
was.
what was.
The
The ratio
ratio 2};
can be
What is
2~ : 1I;~ can
be written
written as
as n
n:: 2.
2. What
the value
value of 11
the
n ??
Julie
has a
Julie has
square fence that
a square
that encloses
encloses her
her garden.
garden .
She decides
expand her
her garden
decides to expand
garden by making
making
each
10percent
After
each side
side of the
the fence 10
percent longer.
longer. After
this
will
this expansion,
expansion, the
the area
area of Julie’s
Julie 's garden
garden will
have
percent?
have increased
increased by what
what percent?
A) 20%
A)
20°/o
The
the
The price
price of Product
Product X
Xisis 25% greater
greater than
than the
price
The price
Product Z is 25%
price of Product
Product Y.
Y.The
price of Product
less than
the price
product Y.
What is the
the ratio
ratio
than the
price of product
Y.What
of the
the price
of
Product
X
to
the
price
of
price Product
the price
Product Z?
Z?
Product
B) 21%
8)
22%
C) 22°/o
D) 25%
A
A)) 3 : 2
B)
8) 4:
4:33
A right
hass a base
radius of rand
r and aa
right circular
circular cone
cone ha
base radius
height of 11.
If
the
radius
is
decreased
by
20
height
11
. If the radius
decreased
20
percent
the height
10
percent and
and the
height is increased
increa sed by 10
percent,
the following
is the
resulting
percent , which
which of the
following is
the resulting
percent
volume of the
cone?
percent change
change in the
the volume
the cone?
C
C)) 5 : 2
D
D)) 5 ::33
A)
A) 10°/o
10% decrease
decrease
B)
8) 12% decrease
decrease
C) 18.4%
18.4% decrease
decrease
D)
decrease
D) 29.6% decrease
41
41
CHAPTER 5
RATIO & PROPORTION
B
45°
h
A
A
s\/§
s
b2
D
D
40
5
45°
I
S
s
The
The area
area of the
the trapezoid
trapezoid above
above can
can be
be found
found
In
sides
ln the
the triangle
triangle above,
above, the
the lengths
lengths of
of the
the sides
relate to one
relate
another
as
shown.
If
a
n
e
w
one another as shown. If a new triangle
triangle
is created
such that
that the
the area
area of
of
created by
by decreasing
decreasing ss such
the nnew
the
e w triangle
the original
original area,
triangle is 64
64 percent
percent of
of the
area,
s must
5
must have
have been
been decreased
decreased by
by what
what percent?
percent?
using
If lengths
BC and
and
using the
the formula
formula ~%(b1
(bi + b2)h.
b2)l1.If
lengths BC
AD are
is doubled,
are halved
halved and
and the
the height
height is
doubled, how
how
would the
would
the area
area of the
the trapezoid
trapezoid change?
change?
A) The
50 percent.
The area
area would
would be
be increased
increa sed by
by 50
percent.
B)
B) The
The area
area would
would stay
stay the
the same.
same.
A)
A) 8°/o
8%
B) 20%
C) The
25 percent.
percent.
The area
area would
would be
be decreased
decreased by
by 25
D) The
50 percent.
The area
area would
would be
be decreased
decreased by
by 50
percent.
C) 25%
D) 30%
V
T ---------Questions 10-11
Questions
10-11 refer to
the following
to the
following
information.
information.
----------
Calvin has
has a
a sphere
Calvin
sphere that
is four
that is
four times
times bigger
bigger
than
Kevin has
in terms
The
than the
the one
one Kevin
has in
terms of
of volume.
volume. The
radius
radius of Calvin’s
Calvin's sphere
sphere is how
how many
many times
times
greater in length
greater
length than
than the
the radius
radius of Kevin’s
Kevin's
sphere (rounded
(rounded to the
sphere
nearest hundredth)?
the nearest
hundredth)?
L = 4nd2b
The
star each
The total
total amount
amount of energy
energy emitted
emitted by
by a
a star
each
second
luminosity L,
related to
second is
is called
called its
its luminosity
L, which
which is
is related
to
d, its
1),
its
its distance
distance (meters)
(meters) away
away from
from Earth,
Earth, and
and b, its
brightness
brighb1ess measured
measured in
in watts
watts per
per square
square meter,
meter, by
by
the
the formula
formula above.
above.
A) 1.44
B) 1.59
1.59
C) 1.67
D) 2.00
If one star is three times as far away from Earth
as another, and twice as bright, its luminosity is
how many time s greater than that of the other
star?
A
A)) 8
B) 9
C
6
C)) 116
D
8
D)) 118
42
42
THE COLLEGE
COLLEGE PANDA
PANDA
Astronomers see two equally bright stars, Star A
and Star B, in the night sky, but the luminosity of
Star A is one-ninth the luminosity of Star B. The
distance of Star A from Earth is what fraction of
the distance of Star B from Earth?
If
If the
the ratio
ratio of y
y:: 2.4 is
is equivalent
equivalent to
to 2.7
2.7 :: 3.6, what
what
the value
is the
value ofy
of y??
3
A);
A) 2
4
B);
B) 3
C) 7
C);
3
9
D);
D) 5
1
Alfi
A) 27
1
B) 9
1
C) 3
2
D) 3
Box A weighs
42 pounds
and Box B
weighs 30
30
weighs 42
pounds and
B weighs
pounds . The
pounds.
The ratio
ratio of the
A to
the weights
weights of
of Box A
to Box
equal to the
B is equal
the ratio
ratio of the
the weights
weights of
of Box C
C to
to
Box D. If
weigh a
of 180
If Box C and
and Box D weigh
a total
total of
pounds, what
pounds,
what is the
the weight
weight of Box C,
C, in
in pounds?
pounds?
A
The student
student body
The
after-school program
program
body at an
an after-school
consists only
only of 6th
consists
7th graders,
and 8th
8th
6th graders,
graders, 7th
graders, and
graders. The
The ratio
ratio of 6th
graders.
is
6th graders
graders to 8th
8th graders
graders is
17: 28. If a total
total of 110 students
17:
the
students attend
attend the
program , n
11 of whom
whom are
program,
graders, what
are 7th
7th graders,
what is aa
possible
value of n
?
possible value
11?
A
0
A)) 550
75
B) 75
C) 105
D) 130
bookstore ordered
A bookstore
10
ordered an
an initial
initial shipment
shipment of 10
paperback copies
paperback
hardcover copies
of aa
copies and
and 4 hardcover
copies of
newly published
published book.
newly
store m
u s t order
book. The
The store
must
order aa
second shipment
second
shipment with
with the
the same
same ratio
ratio of
of
paperback and
and hardcover
paperback
hardcover copies
copies as
as the
the initial
initial
shipment. If the
shipment.
the store
50 hardcover
store orders
orders 50
hardcover copies
copies
the book
book for the
of the
the second
second shipment,
shipment, how
how many
many
paperback copies
paperback
store order?
copies should
should the
the store
order?
43
43
Expressions
Expressions
33m ~ k
x2 +
variables. Both x2
and variables.
combinations of numbers
expressions are
Algebraic expressions
are just
just combinations
numbers and
+ y and
and m - k are
are examples
examples
Algebraic
2
with
deal with
you to deal
allow you
that will allow
techniques that
fundamental techniques
we'll cover
of expressions.
this chapter,
chapter, we’ll
cover some
some fundamental
expressions. In this
and effectively.
expressions quickly
involving expressions
questions
quickly and
effectively.
questions involving
Terms
1.
Like Terms
Combining Like
1. Combining
they
look like
that look
together that
terms together
putting terms
avoid is putting
the most
terms, the
When combining
When
combining like
like terms,
most important
important mistake
mistake to avoid
like they
combine a + ab to
can you
b3, nor
make £73,
b2 + b to make
' t. For example
can
can't.
example,, you
you cannot
cannot combine
combine b2+
n o r can
you combine
but can
together but
go together
can go
match.
variables have
the variables
add or subtract
make
subtract,, the
have to completely
completely match.
2ab. To add
make 21111.
EXAMPLE 1:
EXAMPLE
2
2 2
10b2 )
5a2 b2 - 10172)
) -‐ (a
20132 -‐ 3a
3a2b2
4172)
(a22 +
+ 5a2b2
b -‐ 4b
2(2a
the expression
equivalent to the
following is equivalent
the following
Which
expression above?
above?
Which of the
2 b2 -‐ 1st
18b 2
3a2 -‐ lla
B)
B) 3a2
11a2b2
A) -‐6a2b2
6a2 b2
2(2a2
2 (2a 2
2 2 -‐
b
3a12172
-7 31
2
2
41,2)
<a2
) -‐ (a
4b
2 b 2 + 2b 2
3a 2 -‐ lla
C) 3,12
11a2b2
+ 2172
2 2
b -‐
+
+ 5a
5a2b2
2
10b
10172))
IT§].
Answer (C) l.
Answer
44
D) 5a2 + 2a2b2 + 21:2
2 2
10b 2
b +
a 2 -‐ Sa
8b 2 -7 112
8b2
5a2b2
+1on2
4a 2 =
: 4a2
6a 2 b2 -‐
6112172
: 3112
3a 2 ‐=
2 2 + 2b22
11112172
b + 2b
11a
THE
THE COLLEGE
COLLEGE PANDA
PANDA
..
v0
2. Expansion
Expansion and
2.
and Factoring
Factoring
J
EXAMPLE
M L ] ;2
2:
:
~ -.
.r
-
4)(2:c +
+ 3)
2(x -‐ 4){2x
Which
Which of the
the following
following is equivalent
equivalent to the
expression above?
the expression
above?
A) ~ 2 -‐ lOx
A)4x2
10x‐24
- 24
B)4x2+10x‐24
B)
4x2 + lOx - 24
C)4x2+10x+24
C) 4x2 + lOx + 24
D)8x2
0 x -‐ 224
4
D) 8x 2 ‐- 220x
Some people
people like to expand
Some
expand using
using a method
called FOIL (first,
method called
(first, outer,
outer, inner, last).
last). If
If you
you haven't
haven't heard
heard of it,
that'
it's the same
that’ss totally
totally fine. After all,
all, it’s
asdistributing
same thing
thing as
distributing each
each term.
term. First,
First, we
we distribute
distribute the ”2.”
"2."
2(x -‐ 4)(2x
+ 3)
4)( 2x + 3) =
= (2x ~- 8)(2x
8)( 2x +
Notice that
that itit applies
applie s to just
one of the
Notice
just one
t w o factors
factors.. Either
Either one
but N
O T both.
the two
one is fine, but
NOT
both.
(2x -‐ 8)(
2x + 3) =
(2x
8)(2.r
: 4x2
4x 2 + 6x ‐- 16x ‐- 24
=
4x 2 ‐: 4x2
10x -‐ 24
10x
[ED.
Answer (A) .
Answer
Now
factoring and
N o w when
when it comes to factoring
and expansion,
expansion, there
there are
are several
several key formulas
formulas you
you should
should know:
know:
.• (a+b)2
(a + b)2 =a2+2ab+b2
= a2 + 2ab+ b2
0
(a‐b)2
=az‐2ab+b2
• (a
- b) 2 =
a2 - 2ab+ b2
0
(a+b)(a‐-b)
• 112‐1)2
a2 - b2= (a
+ b)(a - b)
Memorize
Memorize these
They show
show up
very often.
these forwards
forwards and
and backwards.
backwards. They
up very
often.
EXAMPLE
EXAMPLE 3:
3: Which
Which of the
is equivalent
equivalent to
the following
following is
to 414
4.x4‐- 9y2
9y2 ?
?
A) (2x2
(2x2 +
9y)(2x2 -‐ y)
A)
+ 9y)(2x2
B) (4x2
(4x2 + 3y)
(x2 -‐ 3y)
8)
syxxz
(x 2 + 3y)(4x2
3y)(4x2 -- sy)
3y)
C) (x2
D)
(2x2 + 3mm2
3y)(2x2 -‐ 3y)
D) (2x2
By)
Part
makes for a
score is pattern
Part of what
what makes
a top
top SAT score
pattern recognition.
recognition . Once
Once you've
you 've done
done enough
enough practice,
practice, you
you should
should
2
2
be able
asa
difference
of
t
w
o
squares,
a
variation
of
the
a2‐
b2
formula.
The
be
able to recognize
recognize the
the question
question above
above as a difference of two squares, a variation of the a - b formu la. The
SAT will rarely
on
those
formulas
in
a
straightforward
way.
Beon
the
lookout
for
variations
that
rarely test you
you on those formulas in a straightfo rward
Be on the lookout for variations that
match
match the pattern.
pattern . With more
more practice,
practice, you’ll
you 'll get better
better and
and better
better at
at noticing
noticing them.
them .
Using the formu
formula
= (a +
+ b)(a -‐ b), we can
By. Therefore,
Therefore,
Using
la a4122 -‐ b2
b2 =
can see that
that a =
= 2x2
2x 2 and
and b =
= 3y.
4x4
9y2 =
: (2x
(2x22 + 3y)(2x2
4x 4 -‐ 9y2
3y)(2x2 ‐- ay)
3y)
ITelJ
.
Answer
Answer (D) .
45
CHAPTER 6 EXPRESSIONS
CHAPTER
EXPRESSIONS
EXAMPLE
EXAMPLE 4:
4 -‐ 8x
2 2
16x4
8x2y2
+ 3;4
16x
y +
y4
Which
above?
Which of the
the following
following is equivalent
equivalent to
to the
the expression
expression shown
shown above?
A) (41:2 + f ) ’
4
B)
(2x -‐ y)
B) (2:
y)4
2
C)
C) (2x
(2x +
+ y)2(2x
y)2(2r -‐ y)
y)2
D) (4x + y)2(x ‐ y)2
2 -‐ 2ab
2 (in reverse),
Using the
can see that
that a =
= 4x2
4x 2 and
y2.Therefore,
Therefore,
Using
the formula
formula (a -‐ b)2 =
= aa2
Zab +
+ bb2
reverse), we
we can
and b =
= 3/2.
2
16x44 -‐ 8x
8x2y2
y4 := (4x2
yz)2
16x
y2+ y4
(4x 2 -‐ y2
)2
2 formula to the
This is not
take it one
step further
further and
and apply
apply the
a 2 -‐ bb2
n o t in the
the answer
answer choices
choices.. We have
have to take
one step
the a2
formula
the
expression inside
the parentheses
expression
inside the
parentheses..
2
2
2
(4x
(2x++yy)
( “ z-‐ y2
f f)2
: l=a [(2x
x ++
wy)(
a x2x‐-wy fi)] ==( u
f a(2x
x ‐- wy)2
2
Answer
(C) .
Answer [@]
3.
Combining Fractions
3. Combining
When
When you're
you’re adding
adding simple
simple fractions,
fractions,
11 + 11
3 4
-3 + 4-
the
do this
this so
so that
that we
can get
a common
the first step
step is to find the
the least
least common
common multiple
multiple of the
the denominators
denominators.. We
Wedo
we can
get a
common
denominator. In aa lot
cases, it's
as it is here,
here, 3 x 4 =
denominator.
lot of cases,
it’s just
just the
the product
product of the
the denominators,
denominators, as
z 12.
4
1
5 +
‘ fi
3
7
fi ‐ fi
Now
're adding
denominator, the
the idea
idea is the
the same.
same .
N o w when
when we
we’re
adding fractions
fractions with
with expressions
expressions in the
the denominator,
EXAMPLE5:
EXAMPLES:
1
2
1
2
x+2 + x - 2
x+2+x‐2
Which
equivalent to
Which of the
the following
following is
is equivalent
to the
the expression
expression above?
above?
3x -‐ 22
3x
A)
A ) (x+2)(x
(x+2)(x' -‐ 22))
3x +
+ 22
3x
1)” (x
( x+ 22)(x
) ( x-‐ 22))
B
C)
3
3
C ) (x
( x+ 22)(x
) ( x-‐ 22))
2
2
D ) ((xx++ 22)(x
) ( x-‐ 22))
D)
The common
denominator s: (x + 2)(
multiply
common denominator
denominator is just
just the product
product of the
the two
t w o denominators:
2)(xx -‐ 2). So
So now
n o w we multiply
't have:
have :
the top and
and bottom
bottom of each
each fraction
fraction by the
the factor the
theyy don
don’t
--
11
+
x+
+2
22 _ 11 ix‐2
x- 2
22 .x+2_
x+2
2(x
2(x
x -‐ 2
+
2 ( x++ 22))
_ ((x
x ‐-22)
) ++2--,--0( x ++--'-22))
= -. -+ -. - - = ----+ -----'---= -'----'---,x -‐ 22 _ xx ++ 22 x -‐ 2 x -‐ Z2 x + 2 _ ( x(x++22)(x
) ( x -‐ 22)) ((xx++ 2)(x
2 ) ( x- ‐ 2)
2 ) _ ((xx++ 2)(
2 )x( x- ‐2)2 )
_
3 x++ 2
3x
+ --
‐ ( x(x++22)(
) ( xx -‐ 22))
[ill).
Answer ( B ) .
Answer
46
THE COLLEGE
COLLEGE PANDA
PANDA
4.
4. Flipping
Flipping (Dividing)
(Dividing) Fractions
What's
between+
What’s the
the difference
difference between
1
-
1
- ??
2
2
3
3
and
and -
!2
The
where the
fraction line
divided by 3. The
divided by ~The difference
difference is where
the longer
longer fraction
line is. The
The first is %divided
The second
second is 1 divided
g.
3
They
're not
They’re
not the
the same
same..
11
_ !1 -'-3
; =_ !1 X !1=- !1
2·
2 3
6
3 ‐2'3‐2x3‐6
_i_
_L=
3
1
. 2
--=2
1 -'--=
. 3
3
1 X-=2
3
2
3
The
The shortcut
shortcut is to flip the
the fraction
fraction that
that is in the
the denominator
denominator.. So,
a
- g
b-
_ ac
‐ Fb
C
c
If the
occurs :
If
the fraction
fraction is in the
the numerator,
numerator, then
then the
the following
following occurs:
a
_ b
5_= _1 a
c _ bc
be
EXAMPLE
EXAMPLE 6: If
If x > 1,
1, which
which of the
the following
following is equivalent
equivalent to
?
‐‐1‐xT1 ?
1
X
x --‐ 1 + x + 1
2x22
2x
__
A) (x - l)(x + 1)
A)(x‐1)(x+1)
B)
22
_______
B’(x‐1)(x+1)
(x - l)(x + l)
C)
x(x‐1)(x+1)
x(x
- l)(x + 1)
_______
2
2
C)
D __
)(x+1)
D) ((xx-‐ ll)(x+l)
2
)
2
First, combine
fractions on the
common denominator
denominator (x -‐ 1)(x +
First,
combine the
the two
t w o fractions
the bottom
bottom with
with the
the common
+ 1).
11
1l
X
_
x +1
-x-- 1 + -x +-1 = -(x--- 1)-(x-+~1)
X
x -‐
1l
2Xx
2
x--- 1-)-(x_+_l_) = -(
x‐1+x+1“'u‐1nx+u + -(u‐1Mx+n
ux--‐ 11-)-(
x xx_+_
+l_n)
Next,
Next, substitute
substitute this back
back in and
and flip it.
+11))
x
_ x (x
( x-- 1l ))(x
( x + 11)) _ ((xx -‐ 1l ))(x
(x+
X
‐ ‐ 22xx ‐ _ _ ‐ ‐ 22x§ _ _ ‘ ‐ ‐ 22 _ ‑
( x -‐ 11)(x
) ( x + 1l )
(x
Answer ~(D) .
Answer
47
CHAPTER 6 EXPRESSIONS
CHAPTER
EXPRESSIONS
5. Splitting
Splitting fractions
fractions
5.
30
EXAMPLE 7: Which
EXAMPLE
Which of the
followingisequivalent
to 306+
the following
is equivalent to
: C7
c?
+c
A)
A S5) +c
?
6
10+c
B)
10
+c
B)‐2‐‐
2
C
) 5 +c
+c
C)S
We can split
Wecan
split the
the fraction
fraction into
into two:
two :
ic
D)5+g
0)5
+
30+c_?1)+£_5+£
6
_ 6
6 _
6
The
answer is ~( D ) .- This
The answer
just the
the reverse
adding fractions.
fractions.
This is just
reverse of adding
Note that
Note
that while
while you
you can
can split
split up the
the numerators
fractions, you
so with
numerators of fractions,
you cannot
cannot do so
with denominators.
denominators . So,
3
3
x +y
X
_
7g_
-i=-+x+y
x
3
_
y
3
!J1
you cannot
cannot break
break up a fraction
fraction like
In fact, you
like -x i -y any
further..
any further
x +y
48
COLLEGE PANDA
THE COLLEGE
PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 284.
should N
A calculator should
O T be
NOT
be used
used on the
following questions.
following
questions.
4 + 8x
Which of the
Which
the following
following is equivalent
equivalent to
to 4 ~:x
12x
forxX76
for
/; O
0??
Which of the
the following
following is equivalent
Which
equivalent to
2y +
6x23]
+ 6ch2
6x
6xy 2 ?
1 ++ Bx
8x
3x
4 + 2x
B)
B)
3x
1
1 + 22xx
C)
3x
A)
A)
A)
A) 6xy(x
6xy (x + y)
B) 12xy(x +
+ y)
2
6x y2(y + x)
C) 6x2y2(y
3y 3
D) 12x
12x3y3
D
D)) 1l
0, then 2
!+
~ is equivalent
equivalent to which
lIff a ,f;
76O,then
+ 2
which of the
the
a
Which of the
the following
following is equivalent
Which
3x44 -‐ 3
3 ??
equivalent to
to 3x
4
following?
following?
A)
3(x 2 + 1)2
A)Mx2+1V
3 + 4a
4a
4a
4 ++33a
a
B)
4a
B) 3(x2
1)22
3(x2 - 1)
A)
A)
C)
D)
C)
C)
D)
D)
3 ( x 3 -‐
30:
1”(35+
) (X + 1)
1)
3(x2 +1)(x+1)(x
+ l )(x + l )(x -‐ 1)
3(X
1)
7
4a
4
(x +1)2
+2(x
+1)(y
1)+
( y ++ 1)
1)22
+ 1)2 +
2(x +
l )(y +
+ 1)
+ (y
a+4
Which of the
Which
the following
following is equivalent
equivalent to
to the
the
expression shown
expression
shown above?
above?
A)
(x ++ yy ++1)2
A) (x
1)2
Which of the following is equivalent to
(x 2 + y) (y + z) ?
A) x 2z + 1/
+ 2)2
B) (x
(x +
+y +
2)2
C)) ( (x
x ++yy)2
F ++22
+ 1/Z
D) (x + y)2
y )2 -‐ x -‐ y
B) x 2y + x 2z + y 2 + yz
C) x2y + y2 + x2z
D) x2 + x 2 z + y2 + yz
49
CHAPTER 6 EXPRESSIONS
EXPRESSIONS
CHAPTER
If
yé0
yéy,
If y I0 and
and xx Iy, which
which of the
the following
following is
is
2
xy - x
equivalent to xy ‐ x;2 ?
equivalent
?
A)‘i‘
A)
The expression 8x 2 - ~y 2 can be written in the
form
where c is aapositive
form 8(x -~ cy)(x
cy)(x +
+ cy), where
positive
xxyy -" yy
constant. What
What is the
constant.
value of c ?
the value
_ 'f_
1
A)
A) 16
X
B) 'f_
i
X
1
B) 8
xX
y
1
C) C)
C) g-
4
xX
y
D)
D) 7
D)
which of the following
If x > 1, which
following is equivalent
equivalent to
2
xx2(x
(X
1
-----,----1
-=?7
x - 1 x +S
--+-2
3
x‐1+x+5'
C)
D)
D)
2) +
+44
+ 2)(x
2) ( X -‐ 2)
Which of the
the following
following is equivalent
Which
equivalent to the
the
expression above?
expression
above?
A) (x2
(x2 ‐- 2)2
2)2
A)
B)
B) (x2
(x 2 +
+ 2)2
2) 2
C)
C) (x
(x --1)2(x
1) 2 (x + 2)2
2) 2
D)
D) (x
(x +1)2(x
+ 1)2 (x -‐ 2)2
2) 2
A)
5):
Sx +
+ 77
A)
B)
B)
~
6
6
2x + 4
6
5x + 7
1
30x + 42
1
2+ X
-- 1
2- X
If x I76O,
equivalent to
to
0, which
which of the following
following is equivalent
the given
given expression?
the
expression?
2x -‐ 1
2x + 1
2x + 1
B)
B)
2x -‐ 1
A)
A)
4x22 -‐ 1
4x
x2
x2
D
D)) -‐ 1
C)
50
THE
THE COLLEGE
COLLEGE PANDA
PANDA
_\
A
calculatoris allowed on the following
A calcu
I‑
lator is allowed on the following
quest
ions.
questions.
I
‐
“
):
X
2
3x
+ 8x2
8x 2 ‐- 4x
3x33 +
7x2-11x - 7
712‐11x‐7
Which of
of the
following is the
the sum
Which
the following
sum of
of the
the two
two
polynomials above?
above?
polynomials
”
X
A)
A) _L
- x- 2
x‐2
rX
B)
2(x
B) -_ T
{f -
C)
C) 10x
10x55 -‐ 7x -‐ 7
2
4 + 3x 3
D)
D) 1Sx
15x4
+ 3x3 -‐ 15x
15x2 -‐ 7
D)
3x
D) 2(x - 2)
2(x ‐ 2)
Whic
h of
ing is
equivalent to
Which
of the
the follow
following
is equivalent
to the
the
expression above?
above?
expression
A) -‐2a\/E
2afa
A)
B) afa
B)
a\/E
C) ) 3a
C
3 -a ‐2/a
2 fi
D)
D) 3a
3a+8\/E
+ BJa
9(2y)
2 2 + 2(6y)
26 22 7
If yyéo,
y f=O,what
what is
is the
the value
If
MW · B(3y)2
?·
value of
H
"
Which of the
the following
Which
following is
is equivalent
equivalent to
to the
the
expression above
above for x 752
=/=2 ?
expression
?
3 + x 2 - 15x A) 3x
7
A)
3x3+x2‐1Sx‐7
3
B) 3x
1Sx2 -‐ lSx
B)
3x3 +
+15x2
15x -- 77
(5a +
+ 3Ja)
WE) -‐ (2a +
+ sJa)
s fi )
x
--+~
--+2(2 X- x)a
‐
x- 2
xX
C) 2(x - 2)
2(x ‐ 2)
3x
2)
2)
Models
ConstructingModels
Constructing
graphs .
expressions, equations
as expressions,
quantities as
real-life quantities
represent real‐life
you to represent
require you
questions require
model questions
Constructing model
Constructing
equations,, and
and graphs.
chapter is specifically focused
but this chapter
book, but
chapter s in this book,
other chapters
can be
type can
Questions
be found
found in several
several other
Question s of this type
Now
exponential). Now
quadratic , exponential).
linear, quadratic,
types (e.g. linear,
pertain to any of the conventional
't pertain
ones that don
on
on the ones
don't
conventional model
model types
be
won 't be
there won't
quite simple,
actually quite
chapter's difficult. Most
mean this chapter’s
doe sn't mean
that doesn’t
Most of the questions
questions are actually
simple, and
and there
in the chapter
leave the rest to you in
and leave
example s and
two examples
We'll just do two
here . We’ll
concept s here.
any nnew
e w concepts
chapter exercise.
i.
.
l
schooll>µys
th.e school
grade - If
each grade.
students in each
grade levels
school, there
a school,
EXAMPLE 1: At a
there are
are a grade
levels with
with b students
If the
buys n
71
EXAMPLE
s ckers
thenumber
gives the
the following
which of the
equally among
stickers
bedistributed
among the
the students,
students, which
following gives
number of stickers
distributed equally
stickers to be
each
receives?
student receives?
each student
ab
ab
A)
A) 7
n
an
B)
B) an
7
b
bn
C) bn
a
rt
D)~El?
D)
C) 7
ab
divide
student s. To find the number
b) =
The school has a
(a) (b)
: ab
abstudents.
number of stickers each
each student
student receives, we divide
a total of (a)(
ITe:IJ.
~: . Answer
student s: ‐”‐.
the number
Answer (D) .
number of students:
number of stickers n by the number
ab
th drained
tank.was
until it was
rate imtil
was pumped
pumped into
into a tank at a constant
constant rate
was foll.
full. The tank
was then
EXAMPLE 2: Water was
amount of
total
ent-the
rep
could
graphs
at
been filled. Which of the
the following
following
could represent the total amount
had been
than it had
rate than
slower rate
a slower
at a
water
the tank
time?
versus time?
tank versus
water in the
A)
B)
C)
....
s
....2
0
s....fil
§ ....
gt
~
s.
§2
as
at
~~
s“
~
s;
i”
l
~~
0~
Time
Trme
D)
.....
0
“6
...Iii
0
l‐i
533
~j
s1
:
Time
Time
Time
Jhne
Water
be represented
and to
to the
right (positive
slope) .
(positive slope).
the right
up and
going up
line going
a line
by a
represented by
should be
tank should
the tank
into the
pu~ped into
be~ng pumped
Water being
Water
slope).. That
That leaves
leaves
(negative slope)
right (negative
the right
to the
and to
down and
going down
line going
by aa line
represented by
be represented
should be
dramed should
bemg drained
Water being
us
answers C and
D. Since
tank was
drained at
at aa slower
slower rate
is ~(D)
answer is
the answer
filled, the
was filled,
than itit was
rate than
was drained
the tank
Since the
and D.
wi~ answers~
~s with
line
o t as
steep as
as the
going up
up..
line going
the line
as steep
1snnot
down is
going down
lme going
52
\
I
i
~it
0~
Time
Tlllle
5;
the
the
THE COLLEGE
THE
COLLEGE PANDA
PANDA
CHAPTEREXERCISE:Answers for this chapter start on page 285.
should N
NOT
A calculator should
O T be
be used
used on the
following questions.
following
questions.
An intemet
charges
internet service
service provider
provider charg
es a
a one
one time
time
setup
setup fee of $100 and
and $50 each
each month
month for
for service.
service.
If c customers
customers join at the
the same
same time
time and
and are
are on
the
of the
the following
following
the service
service for
formm months,
months, which
which of
A carpenter
carpenter lays
lays x bricks
bricks per
per hour
hour for y hours
hours
and then
and
g bricks
then lays
lays ~
bricks per
per hour
hour for 2y more
more
expressions
the total
total amount,
amount, in
expressions represents
represents the
dollars,
provider has
charged these
these
dollars, the
the provider
has charged
customers?
customers?
2
hours.
terms of x and
hours. In terms
and y, how
how many
many bricks
bricks did
did
he lay in total?
he
total?
A) 100C
A)
100c + 50m
A)
A) 23:31
2xy
B) 100c
100i: + 50cm
5
B)
B) gxy
xy
C) 150cm
C) 5xy
Sxy
D) 1
100m
+ 50cm
00m +
2
3
D)
D) gx+3y
2x + 3y
At aa math
team competition,
there are
are m
m schools
math team
competition, there
schools
with
from each
school. The
The host
host
with n students
students from
each school.
school wants
order enough
such
school
wan ts to order
enough pizza
pizza suc
h that
that
there
student. If there are
there are
are 2 slices for each student.
are 8
slices in one
which
the following
following gives
one pizza,
pizza, whic
h of the
gives
the
schooll must
the number
number of pizzas
pizzas the
the host
host schoo
must
order?
order?
mn
nm
A)
8
mn
B)
A cheese vendo r current ly has 175 pounds of
mozzarella avai lable for sale . If each pound of
mozzarella sells for $8.75, which of the following
functions gives the amount of mozzarella M, in
pounds, still available for sale after d dollars
worth has been sold?
d
M (d) = 175 - 8.75
B)
B) M(d)
M (d) =175‐8.75d
= 175 - 8.75d
A)
T4
C)
8.75
C ) M(d)
7 5-‐ -8 ?
C)
M (d) = 1175
d
D)
=175(8.75)‐d
D) M(d)
M (d) =
175(8.75) - d
m+2n
m + 2n
8
8
D)
D) 2mn
A retail store has monthly fixed costs of $3,000
and monthly salary costs of $2,500 for each
emp loyee . If the store hires x emp loyees for an
entire year, which of the following equations
represents the store's total cost c, in dollars, for
the year?
A)
A) cc == 3,000+2,500x
3,000 + 2, 500x
B)
B) c
c==12(3,000
12(3,000 + 2,500x)
2, 500x)
C) cc = 12(3,
12(3,000) + 2,500x
D) cc== 3,000
3,000 + 12(2,500x)
12(2,500x )
53
CONSTRUCTING MODELS
CHAPTER
MODELS
CHAPTER 7 CONSTRUCTING
A calculator is allowed on the following
questions.
A manufacturing plant increases the
temp erature of a chemical compo und by d
degrees Celsius eve ry m minut es. If the
compound has an initial temperature oft
degrees Celsius, which of the following
expressio ns gives its temperature after x
minutes, in degrees Celsius?
A)
A)
8)
3)
biking for 4
commu te by biking
began a 5-mile cormnute
Kaiba began
rest area
the rest
at the
stopped at
area. He stopped
rest area.
a rest
miles to a
walked for the
then walked
and then
for 15minutes
15 minutes and
faster
bikes faster
If Kaiba bikes
commu te. If
remainder of the commute.
remainder
grap hs
following graphs
which of the following
wa lks, which
he walks,
than he
than
commu te?
his commute?
represent his
cou ld represent
could
m x++ t
mx
d
m
mdd++ t
A)
A)
xX
"U
-0
QJ
d
C)
t +C) t+‐d‑
mx
dx
D)
Hi"
D) t +m
Q)
o:i
2
E: 4
2
Ez
w
s.is 11
u
C: -
6"'
-‐
0
D
(I)
0-L---+---+----+-➔
45
30
15
15
30
45
(minutes)
Time (minutes)
8)
B)
bakers to make
emp loys bakers
cupcake store employs
A cupcake
make boxes
and
cupcakes
x
tains
con
box
Each
cupcakes. Each
of cupcakes.
contains cupcakes and
cupcakes
y
produce
to
expected
baker is expected
each baker
produce cupcakes
expressions
following expressions
Which of the following
each day. Which
each
needed for all
boxes needed
number of boxes
gives
all the
the
gives the number
working for 4
bakers working
produced by 3x bakers
cupcakes
cupcakes produced
days?
days?
T:
-0
QJ
o:i
Q)
2
5.
~ ,...._4 .
E?
4
[r=
2_ 3
m- ~
3
2
u E 2
a5
11
s2
QJ ·-
C: -
(I)
6"'
'‐
D
A) 12x2y
12x2y
3
3y
B)7y
8)
4
C)
5
~ ,...._4
F r:~ 33 .
QJ · -
0 -"""'--+---+----
o
15 30
45
Time(minutes)
Time (minu tes)
C)
C)
12x2
‘U
-0
QJ
y
0.0
o:i
2
5
4
~ ,...._4
«EA
r=
P
g~ 33
D) 12y
·;:
éé
z211
s2
~
C: ..::,
U)
-6"'
5
o
0
souvenir
price of one
At
shop for tourists,
one souvenir
touri sts, the price
At a shop
purchased
additional souvenir
dollars . Each additional
is a dollars.
souvenir purchased
Which
percent.
40 percent. Which
discounted by 40
after
after the first is discounted
C,
cost C,
total cost
gives the total
equations gives
of the following
following equations
where
souvenirs, where
purcha sing n souvenirs,
dollars , of purchasing
in dollars,
n>
> 11??
15 30
45
30
Time (minutes)
(minutes)
D)
D)
-0
QJ
o:i
>
<1l ,...._
5
4
r=~ 3
A) CC == aa +
+(n‐1)(O.4a)
(n - 1)(0.4a)
B)
8) C = a + (n -‐ 1)(0.6a)
QJ · -
u E 2
~ - 1
0
6"'
C) C := a + n(0.6a)
n(0.6a)
D)
= 0.6an
D) C =
15
15
30
30
45
45
Time (minutes)
(min ut es)
54
THE COLLEGE
THE
COLLEGE PANDA
PANDA
Mike starts
starts driving
Mike
driving to
to work
work and
and records
records his
his
distance from
from home,
distance
10 minutes.
home, in miles,
miles, every
every 10
minutes.
His distance
distance from
His
from home
home increases
increases slowly
slowly at first
first
due to traffic,
traffic, then
due
more quickly
quickly as
as
then increases
increases more
traffic clears
clears up.
up. Which
Which of
of the
the following
following graphs
graphs
could illustrate
illustrate Mike’s
could
Mike's distance
distance from
from home
home
during his
during
his drive?
drive?
At a video
video game
be
game arcade,
arcade, d dollars
dollars can
can be
exchanged for p tokens.
exchanged
tokens. If
If each
each game
game requires
requires w
w
tokens to
tokens
gives the
to play,
play, which
which of
of the
the following
following gives
the
cost per
cost
per game,
game, in dollars?
dollars?
w
2
A)
A) dp
dP
A)
A)
8)
B)
Q.I
2m
2m
E
25,c- --r----.--r----,-----.-.
20 >--+--~_.,_---+-~------<
§ i 15 f---+---+----+- - ---.------i
J:: :-::: m
Q.I E
10 1--+-----+-~--±-__. m
0
.c
u -
§
0
id
pw
dw
p
F
C) di”
C)
D) d_p
dp
D)
w
5
(J')
10 20 30 40 50 60
Time (minutes)
To prepare
prepare for landing,
landing, a
a plane
plane descends
descends so
so that
that
B)
3)
altitude decreases
its altitude
decreases at a constant
constant rate
rate from
from
24,500 feet to 17,900 feet in 12
12minutes.
Which of
of
minutes. Which
the following
following equations
the
equations gives
gives the
the altitude
altitude A,
A, in
in
feet, of the plane
its descent
descent
plane t minutes
minutes after
after its
began, for0
began,
s 12?
12?
for O fi
:S tt :S
Q.I
Q)
E
0
o
.c
.:
E
E -A
0 ~
J:: =
:-:::
é
a sE
Q.I
u -
@
E
.....
a
25
20
15
10
5
A) A =17,900
= 17,900 ‐- 550t
B
:17
, 9 0 0 + 550:
B)) A =
17,900
550t
0
0 0 100
Q
0 210
0 3200 30
4 0405 50
0 660
0
(J')
24,500
C) A z= 24,
500 ‐- 550t
D) A =
500 +
+ 550t
= 24,
24,500
550t
Time (minutes)
(minutes)
C)
Q.I
O.)
E
c0
25
20
0E... ..!!:! 15
.....
·- 10
Q.I E
8
u -f:
@
5
.....
.<:
.c
A taxicab charges a dollars for the first mile
traveled and b dollars for each additional mile. lf
a
a particular
particular passenger
passenger traveled
traveled more
more than
than one
one
mile during
mile
during a
a ride
ride that
that cost
cost $24, which
which of
of the
the
following
following represents
represents the
the distance,
distance, in
in miles,
miles, the
the
passenger traveled
passenger
traveled during
during the
the ride?
ride?
(J')
E
(JJ
..<e
;
0
,_[
I
i5 0 100
D
0 210
0 3200 30
4 0405 50
0 6600
Time (minutes)
Time
(minutes)
24 - a - b
A)
24‐ba‐b
A)
D)
D)
b
Q.I
B) 24‐ba+b
24 - a + b
E
25
20
E0 ~ 15
J:: :-:::
10
Q.I E
u ;a
5
.....
8)
0
.c
-·
-
I
1
'I
--,-
24 + a - b
C)
24+l:1‐b
C)
b
24 - a
D) 24b‐a
D)
b
(J')
0
b
00 10
10 20
20 30
00
30 40
40 50
50 60
60
Time (minutes)
(minutes)
55
55
CHAPTER 7 CONSTRUCTING
CHAPTER
CONSTRUCTING MODELS
MODELS
Mark started working as an inspector for a large
construction company on June 1, 2019.
According to his contract, his annual salary will
increase by $15,000 on the first day of June each
year. Which of the following graphs could
model Mark 's annual salary, in dollars, x years
after June 1, 2019?
To move
move into
into a new
new studio
studio space,
space, the
themm members
members
club agreed
agreed to split
of an
an art
art club
split the first month’s
month's
rent of r dollars
dollars equally
rent
equally among
among themselves.
themselves. If
If k
of the members
share, which
members fail to pay
pay their
their share,
which of
the following
following represents
represents the additional
the
additional amount,
amount,
in dollars,
dollars, that
that each
members
each of the remaining
remaining members
must pay
pay to cover
cover the first month
must
month’s
rent?
's rent?
rr
111-‐ k
m
kr
B)
B)
m -‐ k
m
kr(m -‐ k)
kr(m
A)
A)
A)
A)
a
E
73
E
b
sro
N
C)
C)
(V)I )
111
111
kr
D)
D)
m(m ‐- k)
76
3
1 2 3 4 5 6
5123456
B)
B)
75
s
T:
E
E‘
N
76
m
a5
f:
5 1 1223344 55 66
C)
E"
s
3
E
E’
2
50°
76
3
E
< 1 12 23 3 44 55 6
D)
D)
'----------+X
1 2 3 4 5 6
56
e
•
Manipulati)Jg
Manipulating &
Solving Equations
On the SAT,
SAT, there
there is aa huge
huge emphasis
emphasis on
on equations.
equations. To get
get these
these types
right, yott
you must
learn how
how
On
types of questions
questions right,
must learn
to isolate
isolate the variables
variables and
and expressions
expressions you
you want.
want. First,
First, we'll
we’ll cover
cover several
several useful
techniques
in
dealing
with
useful techniquJs
dealing with
equations that
that you
you may
may already
already be
be familiar
familiar with.
with.
equations
I
1. Don
Don’t
like terms
1.
't forget to combine like
should be
be ruthless
ruthless in finding
finding like
like terms
terms and
and combining
combining them.
them. Doing
Doing so
so will simplify
simplify things
things and
and allow
allow you
you
You should
figure out
o u t the
the next
next step.
step.
I
to figure
EXAMPLE1:If2(a+b+2c+3d+1)=
EXAMPLE
1: If 2(a + b + 2c + 3d +I) =3a+2b+4c+6d,findthevalueofa.
3a + 2b + 4c + 6d, find the value of a.
same four
four variables
variables are
are on both
both sides
sides of the
the equation,
equation, a,
a,b,c
should tell you
you to distribute
on the
the
The same
b, c and
and d. That
That should
distribute on
side first and
and then
then combine
combine like
l i k e terms
terms.. Sounds
Sounds simple
simple but
but you
you won
won't
how many
many hudents
students forget to
't believe
believe how
left side
do this, especially
especially in the
the middle
middle of aa more
m o r e complex
complex problem
problem..
do
6 + 33dd + 11)) =
2 33aa +
+ 22bb + 44c
c + 66d
d
2((aa++ bb + 22c
The b,
b,c,
and d variables
variables cancel
cancel quite
quite nicely.
The
c, and
211+25+M+fid+223a
+26+Arf+fad
~
+ ~ + ¥ +M + 2 = ~ +~
+¥ + M
: a
IT]=
\
\
\
“51
CHAPTER 8 MANIPULATING
MANIPULATING & SOLVING
SOLVING EQUATIONS
EQUATIONS
CHAPTER
2. Square
Square equations
equations correctly
2.
When squaring
squaring equations
equations to remove
remove aa square
square root,
root, the
the most
most important
thing to remember
you’re not
not
When
important thing
remember is that
that you're
squaring
-y ou're squaring
the entire
squaring individual
individual elements
elements‐you’re
squaring the
entire side
side..
EXAMP1,.'E2:
EXAMPLEZ:
·"'
vab
= a- b
\/a_b=a‐b
If a,>.
a >. 00 and
and b > 0,'.
O,tthe
equation above
is equivalent
equivalent to
to which
which of the
the following?
following?
U
h;eequation
above is
A)ab=az~-b2
B)ab=az+b2
C)2ab=aZ‐b2
2ab = a 2 - b2
C)
D)3ab=az~l-b2
D) 3ab = a2 + b2
·TThe
problem should
h e square
square root
root in the
the problem
should scream
scream to you
you that
that the
the equation
equation should
should be
be squared.
squared. Most
Most students
students know
know
the square
make:
the
square root
root should
should be
be eliminated,
eliminated, but
but here's
here’s the
the common
common mistake
mistake they make:
ab=aZ‐b2
ab
= a2 - b2
They square
When modifying
equations, you
square each
each individual
individual element.
element. However,
However,this
this is WRONG.
WRONG.When
modifying equations,
you must
m u s t apply
apply
any
any given
given operation
operation to the
the entire
entire SIDE, like
like so:
(M)2
b)2
(fa‐bf =
= (a
(a‐- b)2
If it helps,
the operation.
the same
same holds
holds true
helps, wrap
wrap each
each side
side in parentheses
parentheses before
before applying
applying the
operation. By the
the way,
way, the
true for
both sides
all other
other operations,
operations, including
including multiplication
multiplication and
and division
division.. When
When you
you multiply
multiply or divide
divide both
sides of an
an
equation,
parentheses, but
because of the
equation, what
what you're
you’re actually
actually doing
doing is wrapping
wrapping each
each side
side in parentheses,
but because
the distributive
distributive
property, it just
multiplying or
individual element
the same
property,
just so
so happens
happens that
thatmultiplying
or dividing
dividing each
each individual
element gets
gets you
you the
same result.
result. For
For
example,
example, if we
we had
had the
the equation
equation
zy
x + 22 =
and
both sides
doing is
is
and we
we wanted
wanted to
to multiply
multiply both
sides by
by 3,
3, what
what we're
we’re actually
actually doing
3(x + 2)
2) = 3(y)
3(x
which turns
turns out
out to be
be the
the same
same as
as
which
3x ++ 66 =: 3y
3x
3y
Anyway,
back to the
Anyway, back
the problem:
problem:
2
(M)2
=
MW
= (a
(a -‐ b)
17)?
ab:a2‐2ab+b2
ab
= a 2 - 2ab + b2
2
2
3ab =
: aa2
-+- b
b2
3ab
+
answer is~is (D) .
The answer
Another
the square
square root
root is isolated
isolated on
side. For
For example,
Another common
c o m m o n mistake
mistake is squaring
squaring each
each side
side before
before the
on one
one side.
example,
let's
let’s say we
we wanted
wanted to find
find the
the solutions
solutions to the
the following
following equation:
equation:
Vx+5+lzx
We can't
"1" from
can’t square
square each
each side
side right
right away
away to get
get rid
rid of the
the square
square root.
root. We first have
have to move
move the
the ”1”
from the
the left
side to the
theright
side:
side
right side:
x + 5=: xx ‐ 1l
v'x+5
58
58
\~
___
____
_____
_ _ THE
COLLEGE
PANDA
THE
__ C
_O_L_L_E_G_E_E
_'AND
__ :A
_
And n
now
can square
square both
And
o w we can
both sides.
sides .
(Jx
( v x++5)
52f == ((xx ‐- U1)22
xx+5:x2‐2x+1
+ 5 = x 2 - 2x + 1
00=x2‐3x‐4
= x 2 - 3x - 4
0 := ((x
x ‐-44)(x
) ( x ++ 1l))
X
x
==‐ -1 1,4
,4
solutions are
are ‐1
So, the solutions
hold on! We’re
square roots
in the
- 1 and
and 4, but
but hold
We're actually
actually not
not done
done yet.
yet. When
When there
there are
are s~uare
roots in
the
original equation,
equation, we
original
we have
solutions by
testing
each
of
our
values
in
the
original
equation.
have to check
check for false solutions
by testing each of our values in the original equation.
So when
when x = -‐ 11,, the
50
the left hand
hand side
+ 5+
+ 11 =
z 3
The values
match,
side is \/
J -‐11 +
3 and
and the
the right
right hand
hand side
side is
is ‐- 11.. The
values don't
don't match,
so ‐1
- 1 is actually
actually n
so
o t a solution.
= 4,
4, the
the left
left hand
is \/4
not
solution. When
When xx =
hand side
side is
J4 ++ 55 ++ 11 := 44 and
and the
the right
right hand
hand side
side is
is 4.
4.
case, the
the values
In this case,
sides match
so4
values from
from both
both sides
match so
4 is
is a
a solution.
solution.
Why do false solutions
Because squaring
squaring both
both sides
solutions occur?
occur? Because
sides has
has the
the effect
effect of
of turning
turning negative
negative values
values into
into
positive ones,
ones, which
positive
which sometimes
mismatch on both
sides.
If
we
plug
x
=
‐1
into
\/
x
+
5
=
sometimes causes
causes a mismatch
both sides. If we plug x = - 1 into J x + 5 = xx ‐- 11 from
from
above, we
above,
side is
One is
we can
can see
see that
that the
the left hand
hand side
side is
is 2
2 and
and the
the right
right hand
hand side
is -‐ 22.. One
is positive
positive and
and the
the other
other is
is
negative. Once
negative.
we square
square both
sides,
however,
this
distinction
is
lost
since
both
sides
become
4.
Once we
both sides, however, this distinction is lost since both sides become 4.
In summary,
you’re dealing
with square
roots in
square the
the entire
sides, which
summary, when
when you're
dealing with
square roots
in an
an equation,
equation, square
entire sides,
"'lhich may
may require
require
you
you to move
move something
something from
one side
to the
other, and
solutions
by
plugging
your
from one
side to
the other,
and check
check for
for false
false solutions by plugging your results
results into
into
the
the original
original equation.
equation.
I
This
lot to
to watch
watch oout
u t for,
but for
most of
This may
may seem
seem like
like aa lot
for, but
for most
of the
the questions
questions involving
involving this
this type
type of
o:fequation
equation on
on the
the
SAT,
pitfalls by
by plugging
than solving
solving
SAT, you
you can
can avoid
avoid all
all the
the potential
potential pitfalls
plugging in
in the
the answer
answer choices
choices (see
(see tip
tip #8)
#8) rather
ra~er than
the equation
equation algebraically.
algebraically.
3.
o o t equations
3. Square
Square rroot
equations correctly
correctly
Now,
it comes
both sides,
forget the
Now, when
when it
comes to
to taking
taking the
the square
square root
root of
of both
sides, most
most students
students forget
the plus
plus or
or minus
minus( (±i )).. Always
Always
remember
equation such
such as
asx2
z 25
25 has
has two
t w o solutions:
solutions:
remember that
that an
an equation
x2 =
f o z ¢ fi
x = i 5
However,
only applies
taking the
the square
square root
However, this
this only
applies when
when you’re
you're taking
root to
to solve
solve an
an equation.
equation. By
By definition,
definition, square
square
roots
\/§ =
= 3,
: -‐33 is
is not
n o t possible
when
roots always
always refer
refer to
to the
the positive
positive root.
root. So,
So, v'9
3, NOT
NOT 3&3.
± 3. And
And fy'x =
possi~le (except
(except when
working
numbers, which
which we'll
we’ll look
look at
at in
in a
future
chapter).
The
plus
or
minus
is
only
necessary
working with
with non-real
non-real numbers,
a future chapter). The plus or minus ls only necessary
when
is used
solve an
solutions to
the
when_the
the square
square root
root is
used asa
as a tool
tool to
to solve
an equation.
equation. That
That way,
way, we
we get
get all
all the
the possible
possibl1 solutions
to the
equation.
equation.
EXAMPLE
If (x +
is the
the sum
sum of
the two
EXAMPLE3:
3: If
+ 3)2
3) 2 =
= 121,
121, what
what is
of the
two possible
possible values
values of
of x
x??
u+3fi=1m
(x
+ 3) 2 = 121
tJ(xh ++53)22 =
u1
= i±¢✓121
x ++ 33 =±
= i 11
H
xX =
= -‐ 33±i 11
u
Sox could
s u m of
w o possibilities
Sox
could be
be either
either 8
8 or
or -‐ 14.
14. The
The sum
of those
those ttwo
possibilities is
is j ‐6
- 6 I..
59
CHAPTER
MANIPULATTNG &
& SOLVING
SOLVING EQUATIONS
EQUATIONS
CHAPTER 88 MANIPULATING
4.
Cross-multiply when
set equal
equal to
4. Cross-multiply
when fractions
fractions are
are set
to each
each other
other
Whenever
fraction is
another fraction,
fraction,
Whenever a
a fraction
is equal
equal to
to another
a _ E
C
5
b_ d
you can
cross-multiply: ad
ad =
: be.
you
can cross-multiply:
4
10
10
5
3
.
EXAMPLE4:
If 5xx = ‘3‐', what is
is the value of
o f x ??
EXAMPLE4: If
4
10
s, _ 1_0
5X=3
5 _ 3
12x = 50
50
12x
x_ é
x=l
_ 2:
6I
2
1
.
EXAMPLE 5: If
If ,/:--‐ 2 ‐ ‐ -‐ ‐= 0,
!Sthe value of
o f x ??
EXAMPLE
_]___ =
0~what is
xX -‐ 44 xx+
+2
2
1
2- 4
xx2‐
4 -_ Xx + 22 =
_O
0
2 _ 1
xx7-2 -‐ 4 _ Xx +
+2
2) =
z x2
22(x
(x + 2)
x2 -‐ 44
2x
z x2
2x ++44=
x2 -‐ 44
z x2
00 =
x 2 -‐ 2x
2x -‐ 8
8
) ( x + 22))
0O=: ((xx -‐ 44)(x+
x
X
= 4,
4, -‐2
=
2
we plug
these values
values back
back into
into the
the original
equation, we'll
we’ll see
=4
is a
solution but
= ‐2
If we
plug these
original equation,
see that
that x =
4 is
a solution
but x =
- 2 is
is a
a FALSE
solution because
because it causes
causes division
Therefore, the
the answer
answer is [±].
I. As we learned
solutions can
can
solution
division by 0. Therefore,
learned before,
before, false solutions
occur
an equation
has square
but they
can also
also occur
are variab
variables
in the
the denominator
denominator
occur when
when an
equation has
square roots,
roots, but
they can
occur when
when there
there are
les in
of aa fraction.
Though you
won’t see
solutions very
often on
it’s aa good
good practice
always confirm
confirm
fraction. Though
you won't
see false solutions
very often
on the
the SAT, it's
practice to
to always
your
these two
t w o cases.
cases.
your results
results in these
60
THE COLLEGE
COLLEGE PANDA
PANDA
5. Factoring should
be in your toolbox
toolbox
should be
Some
variables that
that are
are tougher
to isolate
isolate.. For
to do
Some equations
equations have
have variables
tougher to
For a
a lot
lot of
of these
these equations,
equations, you
you will
will have
have to
do
some
the
variable
you
want.
some shifting
shifting around
around to factor out
out the variable you want.
EXAMPLE6:
EXAMPLEG:
a
a
b = -=3a+c
3a +e
Which
the following
Which of the
following expressasainterms
expresses a in terms ofbandc?
of b and e ?
bee
b
A)
1 - 3b
A)1‐3b
b
bee
B)
3b + 1
B)3b+1
1 -‐ 33bb
C) 1
C) be
be
3b+1
D) 3b
+1
D) be
be
_
b =-
11
a-
‐3a+c
3a + e
b ( 3 a+e)=
+ c ) : aa
b(3a
3
a b++be=
b c : aa
3ab
b e z aa ‐-S3ab
ab
be=
b
c : aa(1
( 1 ‐- 33b)
b)
be=
bc
be ‐a
--=
a
1 -‐ 33b} ; ‑
what we
we did?
See what
did? We
We expanded
side. Then
Then we
we
expanded everything
everything out
out and
and put
put every
every term containing
containing a
a on
on the
the right
right side.
were able
able to factor oout
were
u t a and
answer is ~(A) -.
and isolate
isolate it. The answer
EXAMPLE7:
EXAMPLE
7:
xx4+3x3+x+3=0
+ 3x3 + x + 3 = 0
4
What is one
one possible
What
possible real
real value
which the
value of x
x for which
the equation
equation above
above is
is true?
true?
4
xx4+3x3+x+3zo
+ 3x3 + x + 3 = 0
x3(x+3)+(x+3)=0
x3 ( X + 3) + (X + 3) = 0
(x+3)(x3+1)=0
(x
+ 3)(x 3 + 1) = 0
x :=‐ -3 30orr -‐ 1
3 from the first two terms, further factoring was possible with the (x + 3) term. How
Once we
we factored
Once
factored out
out x
x3
two terms, further factoring was possible with the (x + 3) term . How
would you
would
you know
do this? Experience.
know to do
Experience .
61
61
CHAPTER 8 MANIPULATING
CHAPTER
MANIPULATTNG & SOLVING
SOLVING EQUATIONS
EQUATIONS
6.
6. Treat complicated
complicated expressions
expressions as
as one
one unit
unit
EXAMPLES:
EXAMPLES:
x
x3+x2+x= -‐‐‐-‐-‐1-’-‘‑
MCI-FE)
Which of
gives m
in terms of x?
of the following
following gives
min
x?
(x4+x3+x2)R
A) _(x4+x3+x2)\/;j§
B) _
x_312
C) ~(x3+x2+x)(x+-‐)
A) m =
” ‘ m ‐(x+-( fixl) g ) ”‘M(x4+xa+xz)(x+;) ' " ‘ fi g
x
/ x ‐ 1‐
x
D) m =
(.
D ) m = (x3
(x3+x2+x)
+ x2 + x) x +
l)
(x+;)
x
Don't
let the
complicated expressions
expressions as
one unit
Don’t let
the big
big and
and complicated
complicated expressions
expressions freak
freak you
you out.
out. Treat these
these complicated
as one
unit
or variable,
like so:
variable, like
B
A=~
A *me
fi
Multiply
Multiply both
both sides
sides by
by m.
m.
B
mA =z -C‐
mA
C
Divide
both sides
sides by
by A.
Divide both
m_1
_ AC
Finally, plug
plug the
the original
expressions back
back in
in..
Finally,
original expressions
1
x x‐ ‑
xJx-.!.
x
nt =
X
3
2
(x + x + x) ( x +
Answer (D) .
Answer~.
62
i)
THE CKALEGE
cpLLEGE PANDA
PANDA
EXAMPLE 9:
EXAMPLE
(x+1)2+5(x+1)‐24=0
(x
+ 1)2 + 5(x + 1) - 24 = 0
If xx > 0,
what real
value of
of xis
x is the
theequafiOn
above·ttrue?
r u e"'""~
? “1
If
0, for what
real value
'E!quation above
Treat (x +
+ 1) as
as one
one unit
unit and
and call it A.
Treat
(x+l)2+5(x+1)‐24:O
(x
+ 1) 2 + S(x + 1) - 24 = 0
2
A2+5A‐24=0
A
+ SA - 24 = 0
( A++ B)(A
8 ) ( A- ‐3)
3 )=: 00
(A
((xx+
+ 11+
+ B)(x
8 ) ( x++l 1- ‐ 3)
3 )=‐ O0
( x + 99)(x
) ( x-‐ 22)) == 0
(x
x == ‐-99oorr 2
Because
the question
question stipula
stipulates
that x > 0,
0, the
the answer
answer is
is I].
I.
Because the
tes that
Be comfortable
comfortable solving
solving for expressions,
expressions, rather than any
variable
7. Be
any one
one variable
EXAMPLE
EXAMPLE 10: If 3x
3x +
+ 9};
9, wha
whatt is
is the
value of x +
+ By?
9y = 9,
the value
3y ?
Get in the
the habit
habit of looking
looking for what
what you
you want
want before
before you
solve for anything
to get
get
you solve
anything specific.
specific. Is
Is there
there any
any way
way to
the answer
the
answer withou
withoutt solving
solving for x and
and y?
Yes! Dividing
Dividing both
sides of
of the
the given
given equation
equation by
gives x +
Yes!
both sides
by 33 gives
+ By:
3y = I.
[I].
EXAMPLE
EXAMPLE 11: If ~
5=
= 3,
is the
the value
value of iY ??
3, what
what is
2X
y
1
A) E
1
B) 5
z
C) ~
C) 3
~3
D) 5
2
D)
Here,
Here, we
we have
have no
no choice
choice but
but to so
solve
the express
expression.
but we
lve for the
ion . We’re
We're given
given x over
over y but
we want
want y over
over x. We
We can
can
flip the
given equation
get
the given
equation to get
I/
1
=-
!,.._
3
X
ix
Then
we can
divide both
both sides
obtain
Then we
can divide
side s by
by 22 to
to ob
tain the % we’re
we're looking
looking for:
for:
63
63
ix=
21x
1.
1
1
= 8“ Answer
Answer (A) .
=
2 3
: ‐3~
[EI].
CHAPTER 8 MANIPULATING
MANIPULATING & SOLVING
CHAPTER
SOLVING EQUATIONS
EQUATIONS
8. In
In some
some cases,
cases, you
may need
need to plug
answer choices
guess and
and check
check
you may
plug in the answer
choices or guess
When you
When
you can’t
”mathematical” way
way to get
get the
the answer,
answer, you
you have
w o options:
options: 1) plug
plug in the
answer
can't find a
a "mathematical"
have ttwo
the answer
choices or 2) guess
choices
Both are
are valid
valid strategies
strategies that
that you
shouldn't
be
afraid
to
use.
Not
only
does aa
guess and
and check.
check. Both
you shouldn't be afraid
use. Not only does
"brute force”
”brute
often tum
turn out
out to be
be quite
quite efficient,
efficient, but,
but, for some
some questions,
questions, it is the
the only
only way
way to get
get the
force" approach
approach often
the
answer .
answer.
EXAMPLE
EXAMPLEIZ:
12:
v22~x=x‐2
What
the set
set of all solutions
solutions to the
the equation
What is the
equation above?
above?
A)
{- 3, 13}
A){‐3’13}
B){3,6}
B){‐3/6}
C) {13}
C){13}
D)
{6}
D){6}
We could
could solve
solve for x by
We
by squaring
squaring both
both sides,
sides, but
but plugging
plugging in the
the values
values from
choices is actually
actually
from the answer
answer choices
much quicker
much
quicker and
and easier.
easier. We
We just
just have
have to see
see which
which of the
the values
(
‐
3
,
6,
or
13)
satisfy
the
equation.
When
values (- 3,
satisfy the equation. When
x=
the left
hand side
= -‐ 33,, the
left hand
side is y'25
m =
z 5 and
and the
right hand
hand side
z -‐ 55,, so
so -‐33 is not in the
the right
side is -‐33 ‐- 2 =
the solution
solution set.
set.
When x =z 6,
hand side
/16 == 4 and
When
6, the
the left
left hand
side is m
and the
the right
right hand
hand side
the solution
side is 6 -‐ 2 == 4, so
so 6 is in the
solution set.
set. At
this point,
point, we can
answer is~
this
can tell the
the answer
is m
based
on the
the available
available choices,
u t let’s
13 just
be sure.
sure.
based on
choices, b
but
let's test
test x =
= 13
just to be
When x =
When
z 13, the
the left hand
\/§ =
: 3 and
and the
right hand
13‐- 2 = 11, so
so 13
13is
solution
hand side
side is ,/9
the right
hand side
side is 13
is not in the
the solution
set.
set.
EXAMPLE
EXAMPLE 13:
2 (x3 -‐- 4) = 4x
xx2(x3
If xxisis an
an integer,
integer, what
what is
is one
one possible
possible solution
solution to
to the
the equation
equation above?
H
above?
Assuming we can't
can’t use
use a
a calculator,
calculator, there
there is no
no easy
easy way
way to solve
hand, and
and there
there are
are no
no
Assuming
solve the
the given
given equation
equation by hand,
answer
like this
this calls
calls for guess
and check.
check. Typically,
Typically, you
you want
want to
to start
start with
with
answer choices
choices to work
work from.
from. A situation
situation like
guess and
small
numbers like
: -‐ l1,0,
, O,1,
small numbers
like xx =
1, and
and 2.
Il won’t
process here
here since
since it’s
obvious what
what you
you need
do. It
It turns
won't work
work through
through the
the guess
guess and
and check
check process
it's obvious
need to
to do.
turns out
out the
the
answer
l.
answer is is
[II.
64
THE COLLEGE
THE
COLLEGE PANDA
PANDA
EXERCISE1: Isolate the variable in bold . Answers for this chapter start on page 287.
l. A=
nr 2
A=7rr2
22. If
If t = gax,
~ax, find
find ax in terms
terms of
oft.t.
2. C ==22nr
7rr
23. If 3x + 6y = 7z, find x + 2y in terms of z.
fig...
+5 =
3. A = ~bh
24. If x
wh
4.. V
V :=I lwh
25
25.. If a;
2b, find 2x
+ 10 in terms
of b.
= a, find 4t in terms of a.
”dz‐fl1 =a,find4tintermsofa.
V=7rr2h
p-h
p
‐Iz
2
2. .
If p + h
p
‑ ·
find 11
6 Ifp+h =z -33, fmdh
2
26..
.V:7rr2h
: a¥2 ++ b§2
7.. c82 =
I
. terms o f r.
1 + 2r = -1 , find t m
27 . If -27.
1- t
2
If lltztrzéfindtintermsofr.
8..V=s3
V = s3
S227trh+27tr2
9.. S
= 2mh + 2m
2
4x411
4x
a -2
C
2
b =d
If
29. If
C
‘ab1 _ £
d
30.
1f2x(x 3 ‐
30.1f2-‘(x3
lO. b _ d
ll.
2.1/ terms
xY =
z, then
then find
find x
z.
28. If x”
: 2,
xz?’
in terms of 2.
x3 -
X2
4 ), what is pin terms
= p(x
p(x55 -‐ x14),
what p in terms of x?
x?
1
-1)
=
m(x + 11)-:
, whatismin
1)
= m(x2
)‐ ‐,w
h a t i s m in
x
x2
b _ d
xz‐xl
2
terms of xx??
terms
y ==m
12. y
mxx +
+bb
"IZ‐y2_y1
2
”M_x3=
whatisnintermsofx?
31. lf Jx+2 l - x3 = 2...,
l,whatisnintermsofx?
5x2 -‐ 3
nx x
5
n
If a(b2 +
what is
a in terms
terms of
of
32. lfa(b2
+ 22)) + c =
= S(c
5(c + 1)3,
l)3,what
isain
b
and cc??
band
m=y‐2_y1
xz‐x1
33. If
If k(x2
k(x 2 +
4) +
+ 4)
+ ky =
z
2 = “22 +
15. v02
= 11 + 2as
a
16.
xandy?
x and y?
X
bE
=_y2l
b_y2
34. If
ax + 3a + x
andb?
and b?
t=27r\/Z
= 2nJ!j
17. t
7x2 +
+33 h . k .
7x2
f
, what
w at 1s
terms of
o
is k m
in terms
22
8
/p+q
18. A z=mm, ✓p+q
I f XX == :X+1,findXintermsonandZ.
19. If
: ~, find X in terms of Y and z.
Y+Z
20. If
If x(y +
+ 2) = y, find
find yin
y in terms
terms of x.
~a = 1a 126
++1] , find
terms of
“E‑
find a in terms
ofbband
and c.
b
2c
21. If
65
65
+ 3 = b, what
is x in term s of a
CHAPTER 8 MANIPULATING
& SOLVING EQUATIONS
CHAPTER EXERCISE:Answers for this chapter start on page 287.
used on the
NOT
should N
A calculator should
O T be used
questions.
following
following questions.
If;=
~m, what is the value of m?
1
A)
6
A) 6
Ifa+b=‐2,then(a+b)3=
If a + b = - 2, then ( a + b)3 =
2
B
_
B) 3
A)) 4
A
) 3
5
C) 3
B) O
0
C)) -‐ 4
C
6
D)) -‐ 8
D
D) 6
?
4) 2 ?
+ 4)2
4)2 = (n +
11 is (n ‐- 4)2
va lue of n
what value
For what
For
IIff 3x
3 x++ 11 =
= -‐8,whatisthevalueof(x+2)3?
8, what is the va lu e of (x + 2) 3 ?
A
A)) -‐ 1
B) l1
C) 8
D) 125
! i,
ac??
of b -‐ ac
the value
what is the
value ofb
If %x
~ x ~g == 1, what
a
‐k 4+ 2 ‐=
‐3
" ,w
here kk i=
¢ ‐- z2,, what
w ha t i is
's k
' etterms
rms
kinm
where
If k
2
C
o
off xX??
A
A)) -‐ 3
A)
12‐2x
12 - 2x
A)
B) 0
xX
C) 2
B) 12+2x
12 + 2x
B)
the
from the
D) It cannot
determined from
be determined
cannot be
given.
information given.
information
xX
xX
C)
C)
12+2x
12 + 2x
D
1 2 x- ‐ 2
D)) 12x
6x -‐ 7?
value of 6x
the value
If
3 , what
7?
what is the
= -‐ 223,
3x -‐ 8 _‐_
If 3x
A
A)) -‐ 5
va lue of
the value
what is the
and x < 0, what
If
36 and
= 36
3) 2 =
If (x ‐- 3)2
x2??
x2
B
1
B)) -‐ 221
C
C)) -‐ 3300
D
7
D)) ‐- 337
66
THE COLLEGE PANDA
r-
f=p<(1+ii)n‐1)
1)
J = p ( (1 +
lIff y >
0 and
> Oand
J
of yy??
value of
the value
(g)3
what isis the
, what
= 31/5,
(~)3
2
an
value J
future value
the future
gives the
above gives
The
f of an
formula above
The formula
the
payment p,
monthly payment
the monthly
on the
based on
annuity based
annuity
p, the
months n.
the number
and the
rate i, and
interest rate
interest
number of months
terms of J,
pin
gives
following
the
Which of the following gives p in terms
Which
f, i,
n??
and n
and
fiJi
A)
(1 + i)" - 1
M
u+nn‐1
((11++ i)i ) " ‐ 11
B
‐ fi ‐
B) )
Ji
11
-
If
fJ ‐- i
C)
2\/x+4
2Jx + 4
3
.
value
the value
1s the
what is
and x > 0, what
=
= 6 and
off X?
o
x?
+ i)" - 1
C) (1(1+i)"‐1
D
fi +
1 ++ i i)
) "11
D)) Ji
+11 ‐- ( (1
.
n1
n
value of -L ??
the value
1s the
what is
2, what
= 2,
If fl- =
2m
211
Zn
2m
A)
1
8
2
20‐fi=§fi+1o
20 - ✓x = 3 ✓x + 10
1
B) 4
equation
the equation
value of xx is the
what value
If x > 0, for what
true?
above true?
above
1
C) -
2
D) 1
x 2 + Sx - 24
=0
0,
< 0,
above and
equation above
the equation
solution of the
If k is a solution
If
and k <
?
lkl
of
|k| ?
what
value
the value
what is the
the value
+ y2 + 16, what
Jx2 +y2
IIff x +y
what is the
value of
+ y = «X2
xxy?
?
l/
67
CHAPTER 8 MANIPULATING & SOLVING EQUATIONS
- 1
xx ++2 2
2X
X -
A calculator is allowed on the following
questions.
2
_4 T
What is the
the solution
the equation
equation above?
What
solution set
set to the
above?
y+2kx=kx2+5
y + 2kx = kx 2 + 5
A)
A) {{‐ 10,0}
10,O}
In the
the equation
equation above,
above, k is a constant.
constant. If
If y := 23
what is the
the va
lue of k?
when x = 3, what
when
value
k?
B)
10, -‐ 44}}
B) {{‐10,
C)
C)
{0,8}
{0,8}
A)) -‐ 6
A
D)) {D
{ 44,8}
,8}
B) 3
C) 6
D) 9
(92 -‐2(%)
15==o,
what isthe
2(i) -‐ 15
0,whatisthe
IIff xx >
> 0and
Oand (if
value
value of xx??
xX = _
x+12
6
X + 12
.'
,what 1s
the value
of -_
If -_
6 =-42 ,whatis
the
value of
x ?7
.
~
6
X
1
A) 3
3
B) ) 2
B
C) 3
C)
D) ) 6
D
2
x - 4x
+3 = 4
x- 1
What
above?
What is
is the
the solution
solution to
to the
the equation
equation above?
d
= a (~)24
Doctors use
rule, shown
above, to
Doctors
use Cowling's
Cowling's rule,
shown above,
to
determine
right dosage
dosage d, in milligrams,
determine the
the right
milligrams, of
medication for aa child
child based
on the
the adult
adult dosage
dosage
medication
based on
a,in
child’s
a, in milligrams,
milligrams, and
and the
the chi
ld 's age
age c,
c, in years.
years .
Ben is a
who is in need
need of a
Ben
a patient
patient who
a certain
certain
medication.
uses Cowling’s
medication . If a
a doctor
doctor uses
Cowling's rule
rule to
prescribe
prescribe Ben
Ben a
a dosage
dosage that
that is half
half the
the adult
adult
dosage,
Ben's age,
dosage , what
what is Ben's
age, in years?
years?
2 4
: 8x
8x44
xx2(x4
(x -‐ 9) =
A) 7
0, for what
what real
real value
value of x is the
the equation
equation
If x > 0,
above true?
true?
above
B) 9
C
C)) 11
D
3
D)) 113
68
THE
PANDA
THE COLLEGE
COLLEGE PANDA
111
3‐1 ,______
V --------'Y
---------
Qu estio ns 23-24 refer to the following
Questions
following
1
-
A)
B)
C)
D)
the figure
figure above,
In the
above, ttwo
w o objects
objects are
are connected
connected by aa
string
which is threaded
threaded through
string which
Using its
through a pulley.
pulley. Using
weight , object
object 2 moves
moves object
object 1 along
weight,
along a flat surface.
surface.
The acceleration
acceleration a of the
the two
two objects
objects can
The
can be
be
determined by the
the following
determined
formula
following formula
A)
A)
B))
B
---'~--'----=
C))
C
D)
where m1
where
object 1 and
and
m, and
and m2 are
are the
the masses
masses of object
object 2,
object
2, respectively,
respectively, in kilograms,
the
kilograms, g is the
acceleration due
due to Earth's
acceleration
Earth's gravity
gravity measured
measured in
.
The acceleration
acceleration would
The
would be
be quadrupled
quadrupled
(multipled by
(multipled
by a
a factor
factor of 4).
A ---------
2y) ‐- 3z
lf 3(x ‐- Zy)
32 := 0,
0, which
which of the
the following
following
expresses x in terms
terms of y and
and z ??
expresses
a
_ m2g
ng -‐ w11
”mg
1g
a=
m
+ m
z
Ill]] +
/112
m
The acceleration
acceleration would
The
would stay
stay the
the same.
same .
The acceleration
acceleration would
The
would be
be halved.
halved .
acceleration would
The acceleration
would be
be doubled.
doubled .
---------
2
-‐ 2 ,, and
and 14
11 is
a constant
constant
ISa
ec2
sec
2y+32
2y + 3z
3
2 y+
+z
2y
y ++22z2
6y+3z
6y + 3z
. .
known
known as
as the
the coefficnent
coefficient of
((xx ++ 1
) ( x- ‐ 2)
2 ) == 77xx -‐ 118
8
l )(x
friction.
friction.
what is
If x is the
the solution
solution to the
the equation
equation above,
above, what
the value
the
18?
value of 7x
7x -‐ 18
?
Which of the following expresses fl in terms of
the other variables?
A)
/I =
A) y:
a(m]
a(m1 + mg)
m2)
"11ng2
m1m2g2
_ a(m]
mz)
a( m1 +
+ m2)
B)) /In=‐
8
"128
lg
m2g -‐ m
m1g
,,-
C) 14Z 111
ng2g -‐¢:7(l:t;
+1112)
a( m1 +
111
2)
C)
-J
If the
the masses
masses of
If
were
of both
both object 1 and
and object
object 2 were
doubled, how
how would
the acceleration
doubled,
would the
wo
acceleration of the
the ttwo
objects be
be affected?
objects
affected?
informa tion .
information.
1
~......._
___
Zfi
2./x :=xx ‐- 33
m1g
Which
represents all the
Which of the
the following
following represents
the
possible
of xx that
satisfy the
possible values
values of
that satisfy
the equation
equation
above?
above?
a(m1 + m2) - m2R
D
1)-z 4
1 1+122;
‐g ‐0m'“ g
D) /1
111,
A) 1 and
and 9
B) 11 and
and 4
C) 4
D) 9
69
CHAPTER 8
MANIPULATING
& SOLVING EQUATIONS
------------=
2
4
x - 6x + 9
V
~
---------
Questions 31-32 refer
Questions
refer to the
the following
following
9
information.
information.
V
v z= pP(l-r)
a‐
Based on
on the
the equation
equation above,
Based
the
above, which
which of the
following could
following
could be
be the
value of x -‐ 33 ??
the value
W1
The value
The
V of a
value V
a car
car depreciates
depreciates over
overt t years
years
according to the
the formula
according
formula above,
above, where
where P is the
the
original price
original
price and
and r is
is the
the annual
annual rate
rate of
of
depreciation.
depreciation.
2
A);
A) 3
3
B);
8) 2
7
C);
C) 3
9
D);
D) 2
Which of the following expresses r in terms of
V,P, and t?
_I~ V
A) r __
mr
= 1
l - “F
B) r _
= 1+
I Vp
B)r‐1+\/‐P
~
Jx - =
10 = f,/x ‐- ✓-fl
i
m
V
C) r = ~
C)r‐V‐P
- 1
1
1n
the equation
equation above,
the value
In the
above, what
what is the
value of
_P
Jx - 10?
VTTE?
_
l/‘r
w
D
D) )r =r lz--l ‐pT
/6
A) x/E
8) ) h
2V2
B
@
c >3./2
a fi
C)
D) v'14
\/1‐4
xy2+ X - y2-
If a car depreciates to a value equal to half its
original price after 5 years, then which of the
following is closest to the car's annual rate of
depreciation?
1=0
A)) 0.13
A
o m
lf the
th e equation
equation above
above is true
If
values of
true for all
all real
real values
what must
must the
the value
value of x be?
y, what
be?
B
o w
8) ) 0.15
C) ) 0.16
C
om
D
D)) o0.22
70
More
More Equation
Equation Solving
Solving
Strategies
Strategies
1n
this chapter,
In this
chapter, we'll
we’ll touch
t w o equation
equation solving
solving strategies
strategies that
are necessary
necessary for certain
certain types
questions
touch on two
that are
types of questions
involving equations.
involving
equations.
1. Matching
Matching coefficients
1.
coefficients
2 + 8x
EXAMPLE 1: If
If (x +
+ a)
a)22 =
z xx2
what is the
the value
EXAMPLE
Bx + b, what
value of b
b??
It’s
to see
anything meaningful
meaningful right
right away
away on
on both
both sides
sides of
the equation.
Solet's
let’s expand
expand the
the left side
side first
It's hard
hard to
see anything
of the
equation. So
and
that takes
takes us
and see ifif that
us anywhere.
anywhere.
,
( x+
+ aa))2 =
: (x
(x+
) ( . \ '+
+ nn)) =
: x 22+
2ax+n2
(x
+aa)(x
+ 2ax
+ a2
So now
n o w we
we have
have
So
7
7
’?
+ 22ax
n . t+
+ a2“ =
z xx‘2 ++8Bx
x+
x. r2‘ +
+ bb
both sides
sides to be
be equal
each other,
other, the
the coefficients
coefficients of each
u s t be
them up.
up.
For both
equal to each
each term
term m
must
be equal.
equal. Let’s
Let's match
match them
x 3 + g x + fi z x2+§x+g
SO,
So,
2a = B
a2 = b
Solving the
the equations,
equations, a11z
and b =
: ~E.
Solving
= 4 and
71
71
CHAPTER 9 MORE
MORE EQUATION
CHAPTER
EQUATIONSOLVING
SOLVING STRATEGIES
STRATEGIES
Another way
Another
this "ma
”matching
coefficients"
way that
that the
th e SAT tests this
tchin g coefficient
s" strategy
strategy is
is to
to phrase
phrase the
th e question
question in
in the
the context
con text
of infinitely
infinitely many solutions
equation (we'
(we’ll
solutions for aa single equation
ll talk
talk aboutinfinitely
about infinitely many
many solutions
solution s for
for a
a system of
of
equations
next chapter).
single equation
the
equations in the
the next
chapter). A single
equation has infinitely
infinitely many
many solutions
solutions only
only when
when both
both sides
sides of
of the
equation are equivalent.
equivalent. For
equation
instance,
For instance,
3x ++ 66 =
: 3x
3x
3x +
+66
has infinitely
has
infinitely many
many solutions
solutions because
because no
no matter
matter what
what the
= 11 is
the value
value of
of x
x is, the
the equation
equation is
is always
always true
true (x
(x =
is
a solution
a
solution,, x = 2 is aa solution,
z 3 is a
solution, ...
. . . ) Notice
Notice that
down
to
O
=
O,
which,
solution, x =
a solution,
that the equation
eq uation boils
boils down to O = 0, which,
again, is always
always true.
true.
again,
Equa tion s like 3x
Equations
3x +
because what
what’s
them in
in the
the first
place?
+ 6 = 3x + 6 are
are a
a bit
bit weird
weird because
's the point
point of
of dealing
dealing with
with them
first place?
course 3x +
Of course
equal to 3x +
+ 6! But keep
keep in
in mind
mind that
that these
o t meant
meant to
be solved;
+ 6 is equal
these equations
equa tions are
are nnot
to be
solved; they’re
they're
meant to demonstrate
the concept
meant
demonstrate the
concept of infinitely
many solutions.
solutions. Let’s
infinitely many
Let's take
take a
a look at
at how
how this
this concept
concept might
might
appear
appear in an
an SAT question.
question.
EXAMPLE
2:
EXAMPLE2:
a(x2
- 2b) =
4x2 - 12
a(x2-2b)
=4x2‐12
In the equation
equation above,
above, a and
and b are constants.
copstants. If the equation
equation has infinitely
infinitely many
what is the
many solutions,
solutions, what
value
b?
value of b?
Just like
le 1, we expand
the left
left side
the coefficients
coefficients so
Just
like in Examp
Example
expand the
side and
and match
match the
so that
sides are equivalent.
that both
both sides
equivalen t.
Only
equiva lent does
infinitely many
Only when
when both
both sides
sides are
are equivalent
does the
the equation
equation have
have infinitely
many solutions.
solutions.
a(x 2 -‐ 2b) =
4x 2 ‐- 12
a(x2
: 4x2
12
ax
ax22 -‐ 2ab
2ab =
: 4x
4x22 -‐ 12
Comparing the
the coefficients,
coefficients, a = 4 and
2ab = -‐12.
12. Now
Comparing
and -‐2ab
Now we
we can
can solve
solve for b.
-‐2ab
2ab == ‐‐12
12
2
-‐2(4)b
2(4) b = -‐ 112
b = - 12 = (TI]
b=“_‐1§:‑
-8
EXAMPLE
EXAMPLE 3:
kx+3(5-2x)
kx+3(5
- 2x) ==15
15
In the
equation above,
above, I:
k is a constant.
constant. If the
the equation
true for
for all
all values
values of x, what
the equation
equation is true
what is the
value of k ?
the value
?
This
question is just
testing you
on the
This question
just another
another way
way of testing
you on
the infinitely
infinitely many
many solutions
so lutions concept.
concept.
kkxx +
+ 3(5
3 ( 5- ‐ 2x
2 x) )=
: 115
5
kkxx+
+1
155-‐ 66xx =
=1155
the right
right side
out the
the x terms
Since the
side is just a constant
constant of 15, we need
need to cancel
cancel out
both
terms on the
the left
left side
side in order
order for both
sides
be equivalent.
easy to see
that kk =
does the
the job. If
If it helps,
sides to be
equivalent. It's
It’s easy
see that
z ~
E] does
side
helps, you
you can
can think
think of the
the right
right side
as
having
a
Ox
term
.
The
end
result
is
th at no
matter what
what the value
value of x is,
is, 15
equa ls 15. Yes, I know
as having a 0x term. The end result that
no matter
15 equals
know these
these
equations are
are weird,
weird, but
but that’s
that's how
yo u get
get infinitely
equations
how you
many solutions.
infinitely many
soluti ons.
72
72
THE COLLEGE
COLLEGE PA
THE
NDA
PANDA
Now
the opposite
opposite of infinitely
N
o w the
infinitely many
solutionss is
is no
an equation
hass no
solutions,
are
many solution
no solutions.
solutions. When
When an
equatio n ha
no solution
s, there
ther e are
no values
that satisfy
satisfy it. To illustrate,
no
values of x that
illustrate, the eq
equation
uation
3 x++ 6 ==33xx ++110
0
3x
has no
solution s because
has
no solutions
because there
there is no
value
that can
can ever
make 3x
equal to
to 3x
3x + 10. The
The equation
equation itself
no va
lue of x that
ever make
3x + 6 equal
itself
is aa contradiction.
contradiction. This is even
even more
more obvious
obvious if we
we subtrac
subtractt 3x from both
sides:: we’re
with 6
z 10, which
both sides
we're left with
6=
which is
is
fundamentally false.
fundamentally
N
o w in the equation
+ 6 = 3x +
+ 10, notice
notice that
that the x terms
side have
the same
same coefficient
coefficient of
of 3,
3, but
but
Now
equation 33:
3x +
terms on each side
have the
the constants
cons tant s of 6 and
and 10 are
are different.
the
equation to have
no
solutions,
the
coefficients
of
the
x
terms
different. For an equation
have
solutions,
coefficients
the x
must be
must
be the same
same on both
both sides,
sides, but
b u t the constants
constants must
must be
be different.
different.
EXAMPLE
EXAMPLE 4:
3cx
Box -‐ 4(x +
+1)
= 2(x ‐- 1)
1) =
1)
equation above
The equation
above has
has no
no solutions,
solutions, and
and c is a constant.
constant. What is the value
value of c ‘?
?
Expanding each
each side,
Expanding
side,
3cx
3 c x- ‐ 4(x
4 ( x++ l1)) = 2(x
2 ( x-‐ 1)
l)
3cx
3 c x- ‐4x
4 x- - 44 =
22
2xx-‐ 2
= 6x ‐- 2
3cx -‐ 4 =
The constants
need to get the
the coefficien
coefficients
to match.
Very simply,
3c =: 66 and
and cc =
: [f].
I.
constan ts are
are different,
different, so
so we just need
ts of
of xx to
match. Very
simp ly, 3c
2. Clearing
Clearing denominators
denominators
1
1
When
you solve
solve an
+ 5xx =
: 10, aa likely
likely first
first step
step is
to get
r i d of
of the
the fractions,
fractions, which
are harder
When you
an equation
equation like 5xx +
is to
get rid
which are
harder
2
3
to work
work with.
with. How
How do
do we
we do
By multiplying
multiplying both
both sides
sides by
times
do that?
that? By
by 6.
6. But
But where
where did
did that
that 6
6 come
come from?
from? 2
2 tim
es
3. So
Sothis
is
what
you're
actually
doing
when
you
multiply
both
sides
by
6:
this what you're actually doing when you multiply both sides by 6:
1
1
1
1
x · (2 · 3) + x · (2 · 3) = 10 • (2 . 3)
2éx-(2-3)+%x-(2~3)=lO-(2-3)
3
£x-(Z-3)+%x~(2-Z)le~(2-3)
1.x
· ('/.· 3) + jx · (2 •,3)= 10 • (2 • 3)
3 x+
+2
z 6600
3x
2xx =
We
i d of the fractions
fractions by clearing
clearing the
the denominators.
denominators. Here’s
the takeaway:
takeaway: we
same thing
We got
got rrid
Here's the
we can
can do
do the
the same
thing even
even
when
are va
variables
denominators..
when there
there are
riables in the denominators
73
73
CHAPTER 9 MORE
CHAPTER
MORE EQUATION
SOLVING STRATEGIES
STRATEGIES
EQUATION SOLVING
EXAMPLES:
EXAMPLES:
‐3 + ‐ ‐5 - == 2
-+-2
xX xx+2
+2
H
a solution
solution to the
If x is
is a
0, what
what is
the value
the equation
equation above
above and
and ix > 0,
is the
value of
of x ?
?
same way
way we
In the
the same
wemultiplied
before,, we
we can
multiply
multiplied by
by 2 ·~3 before
can multipl
y by
by x(x
x(x + 2)
2) here.
here.
3
xg-x(x+2)+x+2-x(x+2)=2-x(x+2)
·x(x + 2) + x +5 2 · x(x + 2) = 2 . x(x + 2)
3
5
g-x(x+2)+xj(z-le/+47=2X(X+2)
.j(x + 2) + .x-rt
•x..(x.-+-2f= 2x(x + 2)
1
3(x+2)+5x=2x2+4x
3(x
+ 2) + Sx = 2x 2 + 4x
3x+6+5x=2x2+4x
3x
+ 6 + Sx = 2x 2 + 4x
0 = 2x 2 - 4x - 6
0:2x2‐4x‐6
0 = x 2 - 2x - 3
Ozxz‐Zx‐B
0 :=( x(x‐ -3 3)(x
) ( x ++1l )
=..
GJ.
x=
l but because x > 0,
z B3 oorr x = -‐1butbecausex
O,x
x=
Here 's one
one final example
example that
that showcases
showcases both
both of the
the strategies
Here’s
strategies in this
thi s chapter.
chapter.
EXAMPLE
6:
EXAMPLEG:
3x + 55 _ -‐6x2+11x+5
3x
6x2 + llx + 5
axx+
(x+l)(ax+2)
xx+1
+1+ a
+2 =
_ (x+1)(ax+2)
equation above,
above, 1:75‐g
x f:. - ~ and
the value
In the
the equation
and a is aa constant.
constant. What
What is the
value of a ?7
a
A) ‐6
-6
B) -‐22
C) 2
D) 6
Let's clear
clear the
the denominators
denominators by multiplying
multipl ying both
Let's
both sides
sides by (x +
+ l1)(ax
)(ax +
+ 2):
3x
55
3x
-‐6x2+11x+5
6x2 + llx + 5
x+1~~~(x+1)(ax+2)+ax+2(x+1)(ax+2)‐mm(x+l)(ax+2)
--·(x+
l )(ax + 2)+-- (x+ l )(ax + 2)= (
l )(
) •(x + l )(ax + 2)
x +l
ax + 2
x + ax + 2
3x
5
3x
5
-‐6x2+11x+5
6x + ll x + 5
m
-M
( a x ax
+ +22)) ++ ~
+ 1 =) ~
M
_ m . ·~
x
~
•.(,x-l-'11(
•-(x(+ x
l ).{ax-+-2T
2
3x(ax
3
x ( a x++ 22)) + 5S(x
( x ++ 1
1)) := -‐6x2
6x 2 + 11
llx x++ 5
2
3ax2+6x+5x+5z
3ax
+ 6x + Sx + 5 = ‐6x2+11x+5
- 6x 2 + 1lx + 5
[ill].
2 terms,
Comparing the
the coefficients
coefficients of the xx2
Comparing
terms, 3a = -‐ 66.. Therefore,
Therefore, a =
= ‐- 22.. Answer
Answer (B) .
74
THE
COLLEGE PANDA
THE COLLEGE
PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 293.
A calculator should NOT be used on the
following questions.
1
2
18x2 - 8
3
2
3
b)
the equation
equation above,
above, a and
are constants.
constants .
In the
and b are
?
Which
the value
Which of the following
following could
could be
be the
value of ab
ab?
6x2 + ~x)
3x) =
: M3
bx2 + ex
cx
(x ~x
ax + bx
30(x3 +
30
= 2(ax + b)(ax -
2
A) 6
A)
B) 9
In the
re constants.
the equation
equation above,
above, a, b, and
and ccaare
constants. If
the equation
true for aJI
the
equation is
is true
all values
values of x, what
what is
the
va lue of a + b + cc?.7
thevalueofa
C)) 112
C
2
D)) 3366
D
xx - ~2(33c-+‐8)‐22
4x
(3x + 8) = 2 ( 2 - ~x)
1
-(‐1)
How many
many solutions
are there
there to the
equation
How
solutions are
the equation
above?
above?
The equation
equation has
no solutions.
solutions.
A) The
has no
2
B) The
The equation
equation has
infini tely many
many so
lution s.
B)
has infinitely
solutions.
C) The
The equation
equation has
has exactly
exactly 11solution.
solution.
C)
D) The
has exactly
solutions.
The equation
equation has
exactly 2 solutions.
ln
the equation
above, a and
and b
bare
ts. If
In the
equation above,
are constan
constants.
If
the
equation has
solutions,
the equation
ha s infinitely
infinitely many
many so
lutions, what
what
is the
value
?
the va
lue of ~g-?
3
A) 4
3
( 3-‐ 22x)
x ) = 1122 -‐ 77xx
3xx + aa(3
4
B) -
If the
In the
the equation
equation above,
above, a is a constant.
constant. If
the
equation
solutions, what
equation has
has no
no solutions,
what is the
the value
value
of aa??
3
C) 6
D ) 112
2
D)
A
A)) -‐ 2
B) 2
C
C)) 4
D) 5
ax - b = 3(2x + l )
In the
the equation
equation above,
above, a and
b are constants
constants.. If
ln
and bare
the
equation
has
no
solution,
which
the
the equation has no solution, which of the
following
be the
values of aa and
and bb ??
following could
could be
the values
If (2):
5) =
= 12x
12x22 + bx
bx -‐ 15
(2x + 3)(ax -‐ 5)
15 for all
all
values
value
values of x, what
what is the
the va
lue of b ??
A ) aa =
z 22 aand
n dbb=
= ~- 3
A)
B
=2a
n dbb=
= 33
B)) aa=
and
C
a n db b=: -‐ 3
C)) aa=: 6 and
A) 6
D ) aa == 6 a
n dbb== 3
D)
and
B) 8
C
0
C)) 110
D
2
D)) 112
75
CHAPTER 9
MORE EQUATION SOLVING STRATEGIES
If (x + 3y) 2 = x 2 + 9y2 + 42, what is the value of
x2y2?
IIff n < Oand
+ 9 = (2x
(2x+n)2,
what is
0 and 4x24»
4x 2 + m
mxx+
+ n)2, what
is
the value o
thevalue
+ n ??
off m +
A) - 15
8)) ‐-‐ 9
B
C)) -‐ 3
C
D)) 112
D
2
1
If ~ + ~ = ~,
what is x in terms of p and y ?
If‐+1=
l,whatisxintermsofpandy?
p
x yy
P
6 x = xx ‐-33x(2n
x ( 2 n-‐ 11))
6x
X
In the
the equation
constant. If
equation above,
above, n is a constant.
If the
the
equation has
equation
solutions, what
what is
has infinitely
infinitely many
many solutions,
the value
value of n ??
the
A) 19‐3!
p- y
A)
3)
fly
8) _EL
p+y
A) -A)
‐523
8) -B)
‐531
C)
C)
(C) D PY
j l
p- y
D) _EL
W
D)
y- p
4
3
5
D)
D) -
3
(x3+kx2‐3)(x‐2)=x4+7x3‐18x2‐3x+6
(x 3 + kx 2 - 3) (x - 2) = x 4 + 7x 3 - 18x 2 - 3x + 6
In the
constant. If
It the
the
the equation
equation above,
above, k is a constant.
equation
values of x, what
is the
equation is true
true for all values
what is
the
value of k?
k?
value
in
ab + a
a
.
If
= g
+ 5 for all
values of b, what
If -ab2b- a =
b+
all values
what is
1s the
the
A
A)) -‐ 99
value
value of a
a??
B)
8) 5
C) 7
D
D)) 9
1
If
1,what
thevalue
x?
If~i -‐ ‐y4‐
- - = 1,
what is the
value o
off x?
X
X - 4
76
THE
THE COLLEGE
PANDA
COLLEGE PANDA
5
ax - b
xx ++33 ' xx -- 22 ‘ (x
( x + 33)(x
) ( x-‐ 22))
2
The equation
equation above
The
above is true
true for
for all
all x > 2,
2, where
where a
and b
bare
constants. What
and
are constants.
What is the
the value
value of a +
+ b ??
A) 7
B) 13
B)
13
C)) 119
C
9
D)) 221
D
1
4
+
2
_
35
35
- 1
--+--=-x -‐ 1l
xX + 1
1 _x2‐1
x2
If x >
the solution
> 1,
1, what
what is the
the equation
solution to the
equation
above?
above?
The equation (2x - b) (7x + b) = 14x 2 - ex - 16
is tme for all values of x, where band care
constants. If b > 0, what is the value of c ?
A) - 20
8) 20
C) 28
D) 36
+ --3!!.__
=3
n‐l+n+1=
n- 1
11+ 1
_ 3_
If n > 0, for what
If
what value
value of
of n
11is
is the
the equation
equation
above true?
above
true?
77
Systems of Equ
Systems
Equations
ations
A system
refers to
more equations
equations that
that deal
deal with
with the
same set
set of
of variables.
variables.
system of equations
equations refers
to 22 or
or more
the same
+ yy =z ‐77
-‐5.t
sx +
2}; =
z -‐ 112
2
-‐ 33xx -‐ 2y
There are
w o main
main ways
ways of solving
solving systems
of 22 equations:
equations: substitution
There
are ttwo
systems of
substitution and
and elimination.
elimination.
Substitution
Substitution
Substitution is all
Substitution
all about
isolating one
variable,
or y,
y, in
in the
way possible
possible.
about isolating
one var
iabl e, either
either xx or
the fastest
fastest way
Taking
the example
we can
can see
see that
it’s easiest
easiest to
to isolate
in the
equation because
because it
it has
has no
Taking the
example above,
above, we
that it's
isolate y
y in
the first
first equation
no
coefficient. Adding
5x to both
both sides,
sides, we
we get
get
coefficient.
Adding Sx
: Sx
5x -‐ 7
y=
We can
can then
then substitute
substitute they
the y in
second equation
equation with
with Sx
5x -‐ 7
We
in the
the second
7 and
and solve
solve from
from there.
there.
7 )=: -~12
-‐ 33xx -‐ 2(5x
2(5x -‐ 7)
12
10x +
+ 14
1 4z
-‐ 33xx -‐ lOx
= ‐- 1122
3x =
: -‐ 226
6
-‐ 113x
z 2
x=
Substituting xx =z 22 back
intoy
: Sx
5x -‐ 7,y
: 5(2)
5(2) -7 77 =
z 3.
3.
Substituting
back into
y=
7, y =
The
z 2,}/
3, which
can be
be denoted
denoted as
as (2,3).
(2, 3).
The solution
solution is x =
2,y z= 3,
which can
78
THE COLLEGE
PANDA
THE
COLLEGE PANDA
Elimination
Elimination
Elimination is
is about
about getting
getting the
the same
same coefficients
coefficients for
for one
one variable
t w o equations
you can
can add
Elimination
variable across
across the
the two
equations so
so that
that you
add
or
subtract the
the equations,
equations, thereby
thereby eliminating
that variable.
variable.
or subtract
eliminating that
Using the
the same
same example,
example, we
we can
can multiply
multiply the
the first
first equation
equation by
so that
that the
y’s have
have the
coefficient (we
Using
by 2
2 so
the y's
the same
same coefficient
(we
don’t worry
worry about
about the
the sign
sign because
because we
we can
can add
add or subtract
subtract the equations).
don't
equations).
0x +
+Z
= -‐ 114
4
-‐ 110x
2yy =
-‐ 33xx -‐ 2y
2 y:= -‐ 112
2
eliminate y, we
we add
add the
equations.
To eliminate
the equations.
2y =
= ‐- 114
4
-‐10x
10x + 2y
-‐ 33xx -‐ 2
2yy =
: -‐ 112
2
2 -‐ 226
6
-‐13x
13x =
Now, we
we can
can see
see that
that x =
= 2. This
This result
result can
can be
be used
used in either
either of the original
equations to solve
solve for y. We’ll
Now,
original equations
We'll pick
pick
the
the first equation.
equation.
+ 22y
y=
: -‐ 114
4
-‐10(2)
10(2) +
2 y=-: ‐ 114
4
-‐ 2200++ 2y
2
=6
2yy =
6
yy =
= 33
And
we got
substitution: x =
2, y z
= 3.
And finally, we
we get
get the
the same
same solution
solution as
aswe
got using
using substitution:
: 2,y
3.
When solving
solving systems
systems of equations,
equations, you
you can
can use
use either
either method,
one of them
will typically
be faster. If
If
When
method, but
but one
them will
typically be
you see a variable
variable with
w i t h no
no coefficient,
coefficient, like
like in -‐ 55xx +
+ y =
z -‐77 above,
substitution is likely
likely the
the best
you
above, substitution
best route.
route . If
you see matching
matching coefficients
coefficients or you
you see that
that it's
it’s easy
easy to get
matching coefficients,
likely the
best
you
get matching
coefficients, elimination
elimination is likely
the best
route.. The
The example
example above
above was
was simple
simple enough
enough for both
both methods
methods to work
work well
(though substitution
was slightly
route
well (though
substitution was
slightly
personal preference.
preference.
faster).
faster). In these
these cases,
cases, it comes
comes down
down to your
your personal
No solutions
solutions
In the
solutions when
when both
the equation
equation are
the previous
previous chapter,
chapter, we
we saw
saw that
that aa single
single equation
equation has
has no
no solutions
both sides
sides of the
are the
the
same except
except for the
the constants.
constants.
same
similar fashion,
fashion, aa system
system of equations
equations has
has no
no solutions
solutions when
the ttwo
w o equations
equations are
are the
the same
except for their
their
In similar
when the
same except
constants. For
For example,
example, the
the system
system
constants.
3 x+
+ 2y
2 y=z 5
3x
3 x++ 2y
2 y== -‐ 4
3x
has
no solutions
solutions since
since the
the different
different constants
constants (5
(5 vs.
vs -‐ 44)) result
result in equations
contradict each
each other.
has no
equations that
that contradict
other . There
There
isn’t an
an x and
and aa y that
that can
can possibly
possibly satisfy
satisfy both
both equations
equations at the same
time.. Note
Note that
the system
system
same time
that the
isn't
3 x+
+2
= 55
3x
2yy =
6 x++4y
4 y=: -‐ 8
6x
also has
has no
no solution.
solution. Why?
Because the
second equation
equation can
divided by 2 to get the
the contradictory
contradictory equation
also
Why? Because
the second
can be
be divided
equation
we had
had before.
before.
we
79
79
CHAPTER
EQUATIONS
CHAPTER 10
10 SYSTEMS
SYSTEMS OF
OFEQUATIONS
EXAMPLE
EXAMPLE 1:
‐ 1 i j=
= 15
15
--‐ax
ax-12y
41: + By =
= -‐22
4x+3y
If the system
system 0f
of equations
equations above
above has
has no
no solution,
solution, what
what is the
the value
value of a ?
11
?
We
must get
get the
the coefficients
coefficients to match
match so
so that
that we
we can
can compare
compare the
w o equations.
do that,
we multiply
the
We must
the ttwo
equations. To do
that, we
multiply the
second
second equation
equation by
by -‐ 44::
-‐ aaxx -‐ 12y
123; =
z 15
-‐16x
16x -‐ 12y
1211 =
= 8
See how
the -12's
match now?
then we
we get
get o
our
contradicting equations
equations
how the
‐12’s match
now? Now
N o w let's
let's compare.
compare. If aa = ~ then
u r two
t w o contradicting
with
with no
no solution.
solution. One
One constant
constant is 15
15 while
while the
the other
other is 8.
8.
Infinite
Infinite solutions
solutions
In
previous chapter,
chapter, we
we learned
infinitely many
when both
both sides
In the
the previous
learned that
that aa single
single equation
equation has
has infinitely
many solutions
solutionswhen
sides of the
the
equation
equation are
are the
the same.
same.
Similarly,
both equations
equations are
are essentially
essentially the
the same:
Similarly, aa system
system of equations
equations has
has infinitely
infinitely many
many solutions
solutions when
when both
same:
3 x+
+ 2y
2 y=z 5
3x
3 x+
+ 2y
2 y=
= 55
3x
(1,
name just
a few. Note
that the
the system
(1, 1),
1), (3,
(3, -‐ 22),
) , (5,
(5, -‐ 55)) are
are all
all solutions
solutions to
to the
the system
system above,
above, to name
just a
Note that
system
6 x++ 4y
4 y=: 110
0
6x
3 x++2y
2 y=: 5
3x
5
also
divided by
by 2 to
to get
get the
They're
also has
has infinitely
infinitely many
many solutions.
solutions. The
The first equation
equation can
can be
be divided
the second
second equation.
equation. They’re
still
still essentially
essentially the
the same
same equation.
equation.
EXAMPLE
2:
EXAMPLE2:
Bx -‐ 5y
531 =
= 8
3x
8
mx -‐ ny = 32
mx
32
In
the system
equations above,
If the
the system
system has
many solutions,
what
In,the
system of equations
above, m and
and n are
are constants.
constants. If
has infinitely
infinitely many
solutions, what
isthevalueofm+n
is
the value of m + n??
Both
the same
there to be
many solutions.
solutions. We
multiply the
the first equation
equation by
Both equations
equations need
need to be
be the
same for there
be infinitely
infinitely many
We multiply
4 to get
hand sides
get the
the right
right hand
sides to match
match::
12x -‐ 20y = 32
12x
mx -‐ ny
71};
mx
=
= 32
32
N o w we can
can clearly
clearly see
see that
that m
m = 12
12 and
and n =
: 20. Therefore
Therefore,, m +
+ n=
: -.
Now
I32 I.
80
THE
COLLEGE PANDA
PANDA
THE COLLEGE
Word
Word problems
problems
You will most
u n into
into a question
asks you
into aa system
system of
most definitely
definitely rrun
question that
that asks
you to
to translate
translate a
a situation
situation into
of equations.
equations.
Here's
Here's a
a classic example:
example:
EXAMPLE
order lunch
lunch from aa restaurant.
restaurant. Each
EXAMPLE 3: A group
group of 30 students
students order
Each student
student gets
gets either
either a
a burger
burger or
or
asalad.
thepriceofasaladis$6.
Ifthegroupspentatotalof$162,how
a salad. Thepriceofaburgeris$5and
The price of a burger is $5 and the
price of a salad is $6. If
the group spent a total of $162, how
many
many students
students ordered
ordered burgers?
burgers?
Let x be
the number
burgers and
can then
then
be the
number of students
students who
who ordered
ordered burgers
and yy be
be the
the number
number who
who ordered
ordered salads.
salads . We
We can
make
w o equations:
make ttwo
equations:
0
x + y z=330
5
62
5xx +66yy ==1162
Make
Make sure
sure you
you completely
completely understand
understand how
how these
these equations
equations were
were made.
made. This
This type
type of
of question
question is
is guaranteed
guaranteed
to be
be on the
the test.
test.
We'll use
We’ll
system. Multiply
Multiply the
the first
and subtract:
use elimination
elimination to solve
solve this system.
first equation
equation by
by 6
6 and
subtract:
6x+6y
6x
+ 6y == 180
5x+6y
: 162
5x
+ 6y =
x=‑
X=~
18 students got
18students
got burgers.
burgers.
More complex
complex systems
systems
You might
might encounter
bit more
encounter systems
systems of equations
equations that
that are
are a
a bit
more complicated
complicated than
than the
the standard
standard ones
ones you’ve
you've seen
seen
above.
these systems,
and some
equation manipulation
w
i
l
l
typically
do
the
trick.
above. For these
systems, substitution
substitution and
some equation
manipulation will typically do the trick .
EXAMPLE4:
EXAMPLE
4:
y + 3x = 0
y+3x
x2 + 2y
2y22 =
= 76
x2
If
If (x,y)
(x,y) is a
solution to the
y?
a solution
the system
system of
of equations
equations above
above and
and y
y > 0,
0, what
what is
is the
the value
value of
of y?
ln
In the
the first equation,
we isolate
isolate y to get
get y =
: -‐ 33x.
x . Plugging
Plugging this
equation, we
this into
into the
the second
second equation,
equation,
2
2(‐3x)2
xx2+
+ 2(3x) 2
: 76
76
=
2 + 2(9x2)
xx2
2(9x 2 )
= 76
76
xx2+1sz2
+ 18x2
= 76
76
2
2
19x2=76
19x
= 76
x2=4
x2 = 4
xX =
: i± 2
IIff xx = 2,
: ‐3(2)
thenyy =
= ‐- 33((-‐ 22)) := 6.
> 0,y
: ~
IE].2, theny
then y =
- 3(2) =
= ‐- 66.. IIff xx = -‐ 22,, then
6. Becausey
Because y >
0, y =
81
81
OF EQUATIONS
SYSTEMS OFEQUATIONS
10 SYSTEMS
CHAPTER 10
CHAPTER
EXAMPLES:
EXAMPLE
5:
xy
+ 23; = 2
xy+2y=2
) - 6= 0
(-1
)2++(x+2)
(_1
x+2 ‐ 6 _ 0
(x+2)
x+2
1
2
1
IYI??
value for |y|
a possible
what is a
above, what
equation above,
the equation
(x, y) is a solution
If (x,
If
solution to the
possible value
substitution
clever substitution
be a clever
might be
there might
that there
hint that
This is a hint
both equations.
+ 2)'s
the (x -+Notice the
Notice
2)'s lying around
around in both
equations. This
equation,
the
Isolating
one.
this
as
complicated
as
problem
a
somewhere,
especially
for
a
problem
ascomplicated
as
this
one.
Isolating
y
in
the
first
equation,
especially
somewhere,
xy + 2y = 22
y(x + 2)
2) =
z 22
_ 2
y _ x+2
1
1
y__
with z.
equation with
second equation
the second
substitute - - in the
can substitute
this form?
want this
would I want
Why would
y_ =
here, 1
From here,
From
z -‐. - . Why
form? So
50 I can
2
x+2
x+ 2
2
this
as this
such as
manipulations such
simplifying manipulations
any simplifying
out for any
an eye
must
questions, you
tougher questions,
these tougher
do these
you do
As you
As
you m
u s t keep
keep an
eye out
one.
one.
Substituting,
we get
Substituting, we
1
‘
+ (- 11 ) - 6 = 0
)2
_1
((m)
x+2
x + 2 +(x+z)‘6=°
’)
)2+ G)- = o
(f
owe‐6:0
6
+ Y-.- 6 = 0
y2
g+g~6zfl
4
2
f+@‐M=O
+ 2y - 24 = 0
y2
((yy+
+ 66)(y
) ( y- ‐ 4)
4 )== 0
Finally,y
z ‐6
|y| can
either /‑6 or 4 /.
be either
can be
and IYI
4, and
or 4,
- 6 or
Finally, y =
even then,
And even
Practice. And
use? Practice.
can use?
you can
"trick" you
substitution or ”trick”
clever substitution
there's a clever
know whether
you know
will you
How
H
o w will
whether there’s
then,
a
without a
done without
be done
designed to be
are designed
questions are
that SAT questions
keep in mind
Just keep
sure. Just
know for sure.
always know
won't always
you won’t
you
mind that
try
and
back
step
a
take
waJI,
a
hitting
or
circles
in
running
you're
like
feel
you
if
So
steps.
of
number
crazy
crazy number steps. 50 you
like you’re running circles
hitting a wall, take a step back and
error.
and error.
trial and
with trial
comfortable with
be comfortable
you must
score, you
perfect score,
a perfect
else. To get a
something
something else.
must be
?
a possible positive
what is apossible
= 10,
and yz =
5, and
= 5,
8, xz =
= 8,
If xy =
EXAMPLE6: If
EXAMPLE
10, what
positive value
value of xyz ?
result
right sides
multiply the right
and multiply
sides, and
the left sides,
Multiply the
equations. Multiply
three equations.
Multiply all three
trick . Multiply
Here 's the trick.
Here’s
sides.. The
The result
is
2y2z2 =
= 8 -· 5 ~10
- 10
xchyzz2
xzyzzz
= 400
x 2y2z2 =
both sides.
Square
o o t both
sides.
Square rroot
= ± v'400
x2y2z2
/W
= HEW)
= i±220
0
xyz =
answer
Since the
value, the
Notice how
get the answer
able to get
were able
we were
how we
/ 20 /. Notice
answer is -.
the answer
positive value,
a positive
asks for a
question asks
the question
z.
x, y, or 2.
values of x,y,
without knowing
individual values
the individual
knowing the
without
82
THE
COLLEGE PANDA
THE COLLEGE
PANDA
Graphs
Graphs
Leaming
our understanding
understanding of systems
Learning aa bit
bit about
about equations
equations and
and their
their graphs
graphs will
will inform
inform our
systems of equations
equations..
The solutions
solutions to a system
system of equations
equations are the intersection
intersection points
points of the graphs
equations. Therefore,
The
graphs of the
the equations.
Therefore ,
the number
points .
number of solutions
solutions to a system
system of equations
equations is equal
equal to the number
number of intersection
intersection points.
Take, for example,
example, the system
system of equations
equations at
at the
the beginning
beginning of this chapter:
chapter:
-‐5x
5x + yy == -‐77
-‐ 33xx -‐ 2};
2y = -‐ 112
2
We
can put
put both
both equations
equations into
into y = mx
mx +
‐+- b
bform
form (we
(we won't
won’t show
show that
that here)
here) and
get the
the following
We can
and graph
graph them
them to get
following
lines
lines..
y
(23)
The
one intersection
point, so
so there
The solution
solution to the system,
system, (2,3), is the
the intersection
intersection point.
point. There
There is only one
intersection point,
there is only
only
one solution
one
solution..
What
What about
about graphs
graphs of systems
systems that
that have
have infinite
infinite solutions
solutions or no solutions?
solutions?
Graphing the
the following
equations contradict
contradict each
Graphing
following system,
system, which
which has
has no
no solution
solution because
because its equations
each other,
other,
y - 22xx=: 1
l
z ‐- 3
y -‐ 22xx =
weget
we
get
y}/
What
points . They're
parallel. Makes
What do you
you notice
notice about
about the
the lines?
lines? They have
have no intersection
intersection points.
They’re parallel.
Makes sense,
sense,
right?
right?
83
CHAPTER
EQUATIONS
CHAPTER 10
10 SYSTEMS
SYSTEMS OF
OFEQUATIONS
And
with infinite
infinite solutions?
And for a system
system with
solutions?
2 y-‐ 4x
4 x=
z 22
2y
y -‐ 22xx = 1
y
It's
just one
it’s two
t w o lines,
lines, but
but because
because they’re
overlap and
It's just
one line! Well, actually
actually it's
they're the
the same
same line,
line, they
they overlap
and intersect
intersect in
in
an
Hence, an
an infinite
infinite number
an infinite
infinite number
number of places.
places. Hence,
number of solutions.
solution s.
EXAMPLE
7: In the
the xy‐plane,
the lines
lines y = 3x
3x -‐ 5
2x+
at the
the point
point (h,k).
(h,k). What
What is
is
EXAMPLE 7:
xy-plane, the
5 and
and y =
= ‐-2x
+1100 intersect
intersect at
the value ofk?
of k ?
thevalue
As mentioned
mentioned earlier,
solutions to aa sys
system
equations are
the graphs
those
ear lier, the
the solutions
tem of equations
are the
the intersection
intersection points
points of the
grap hs of those
eq uations, and
equations,
and vice
vice versa.
versa. So
So to find the point(s)
point(s) where
where two graphs
intersect, solve
solve the system
system consisting
graphs intersect,
consisting of
this problem,
their equations.
equations. 1n
In this
problem, that
that system
system is
: 3x
y=
3x -‐ 5
: ‐2x
y =2x++ 10
10
Substituting the
the first equation
Substituting
equation into
we get
into the
the second,
second, we
get
3 x-‐ 55 =
x ++110
0
3x
=‐-22x
5
x
=
1
5
Sx = 15
x =3
When x =z 3, y = 3(3) -‐ 5 = 4. So
So the
w o lines
lines intersect
intersect at (3,4)
When
the ttwo
(3,4 ) and
and k =z I.
[i].
84
PANDA
COLLEGE PANDA
THE COLLEGE
THE
EXAMPLES:
EXAMPLES:
y=xz‐5x+6
y = x 2 - 5x +6
yy=x+1
=x+1
intersection
an intersection
represents an
y) represents
(x, y)
pair (x,
ordered pair
the ordered
If the
xy-plane. If
the xy-plane.
graphed in the
above is graphed
equations above
system of equations
The system
The
?
value of y ?
possible value
one possible
what is one
equations, what
the ttwo
graphs of the
point
point of the
the graphs
w o equations,
equation
Substituting the first equation
system. Substituting
solve the system.
let's solve
so let’s
points, so
intersection points,
the intersection
are the
the system
solutions to the
The solutions
The
system are
second, we get
into the second,
into
xz‐Sx+6:x+1
x2 - 5x + 6 = x + 1
2
- 6x + 5 = 0
xx2‐6x+5:0
( x-‐ 11)(x-5)=
)(x-5)=0
(x
x ==110orr 55
(1, 2)
intersect at (1,2)
equations intersect
the ttwo
graphs of the
So the graphs
When x =
When
= 1,
1,yy = l1 +
+11 =
: 2. When
When x =
: 5,
5,yy =: 5 +
+11 =
= 6. So
w o equations
[I].
and E.
possible values
which means
(5,6 ), which
and (5,6),
and
means the possible
values of y are[}]
are
and
EXAMPLE9:
EXAMPLE9:
y2=x+3
y2 = x+3
y ==| lxl
xl
yy
does
solutions does
many solutions
How
above. H
shown above.
xy-plane are shown
the xy-plane
graphs in the
and their
equations and
system of ttwo
A system
w o equations
their graphs
o w many
the system
system have?
have?
A)
One
A)One
B)Two
B)
Two
Three
C) Three
D)Four
D) Four
Answer ~-(B) .
solution s. Answer
are ttwo
there are
so there
places so
intersect in ttwo
graphs intersect
Simple.
w o places
w o solutions.
Simple. The graphs
85
85
CHAPTE R 10
CHAPTER
SYSTEMS OFEQUATIONS
10 SYSTEMS
OF EQUATIONS
CHAPTEREXERCISE:Answers for this chapter star t on page 296.
_
A calculator should
O T be
the
should N
NOT
be used
used on the
following questions.
following
questions .
2
+ 5Syy =
2xx+
=2244
x+4y=
x + 4y = 15
15
3 x-‐ 5
= ‐- 1111
3x
Syy =
lX e= ‐1 -B 3yy
If
above,,
If (x,
(x, y)
y ) satisfies
satisfies the
the system
system of equations
equations above
what
what is the
the value
value of x +
+ y ??
A)) 7
A
What
What is the
the solution
so lu tion (x,
(x,yy)) to
to the
the system
sys tem of
equations above?
equations
above?
B) 8
C) 9
D)) 110
D
0
A)) ((-‐ s5,2
A
,2)
B)) ((-‐ z2,1
B
, 1)
C) (1,0)
(1,0)
C)
D) ((4,
- 1)
4,‐1)
D)
_
_
3 x++yy =
= ‐- 22xx + 8
3x
0
-‐ 33xx + 2
2yy = --110
y+
: 2200
+ 22xx =
6
y ==112
2
6xx-‐ 5Sy
If (x, y)
system of equations
equations
1f
y) is a
a solution
sol u tion to the system
above,
w hat is the
above, what
value
the va
lue of xy ??
A)) ‐- 116
A
6
Wh
at is the
the solution
What
solution (x,
system
(x,yy)) to the
the sys
tem of
equations
above?
equations above?
B)) -‐ 8
B
A ) ((-‐ 77,6
,6)
A)
C)) -‐ 4
C
B) (‐6,
B)
(- 6, 7)
D)) 44
D
C)
C) (6,7)
(6, 7)
D)
D) (7,6)
(7,6 )
x+
y z=aax
+ bb
y z= ‐-bbxx
3x -‐ 4y =
= 21
2]
3x
The
w o lines
xy-plane
The equations
equations of ttwo
lines in the
the xy-p
lane are
are
shown
above,
where
a
and
b
are
constants.
If the
the
shown above, where and bare cons tants. If
ttwo
w o lines
value
lines intersect
intersec t at
at (2,8),
(2, 8), what
what is the
the va
lue
of aa?7
=
4x
14
4x -‐ 331
3y = 14
IfIf (x,
(x, y) is a
a solution
solution to the
the system
system of equations
equations
above,
above , what
w hat is the
the value
value of y -‐ x ?
?
A
A)) 2
A
8
A)) ‐- 118
B) 4
B
B)) -‐ 5
C
C)) 6
C) 5
D) 8
D) 8
86
THE COLLEGE
COLLEGE PANDA
THE
PANDA
yy=x2+1
= x +1
z xx -‐ 1l
y=
2
2xx-‐ 4
4yy=
8
2
z8
xX +
+ 22yy = 4
How many
many solutions
are there
the
How
solutions (x,y
(x,y)) are
there to the
system of equations
above?
system
equations above?
y
A) Zero
Zero
A)
One
B) One
C)) Tw
Twoo
C
More than
D) More
than ttwo
wo
2x -‐ 5};
Sy = a
bx + lOy
10y = ‐8
- 8
A system
equations and
their graphs
system of ttwo
w o equations
and their
graphs in
the
man y
the xy-plane
xy-plane are
are shown
shown above
above.. How
H o w many
solutions
solutions does
does the
the system
system have?
have?
ln the
the system
system of equations
equations above,
above, a and
and b
bare
In
are
constants
.
If
the
system
has
infinitely
many
constants. If the system has infinitely many
solutions, what
what is the
the value
solutions,
value of a ??
A) Zero
Zero
A)
One
B) One
C)
Twoo
C ) Tw
A ) -‐ 4
A)
4
D) Three
Three
1
B) 3
4
C) 4
D
6
D)) 116
= y +2
2
-‐ 5Sxx =
2 ( 2 x- ‐1)1 )=: 3 -‐ 33yy
2{2x
What
What is the
the solution
solution (x, y) to the
the system
system of
equations above?
equations
above?
a
x + 2y
2 y=z 5
ax+
5
3 x-‐ 66yy ==2200
3x
A ) ({-‐ 22,8
,8)
A)
the system
above, a
a is a
a constant.
constant.
In the
system of equations
equations above,
the system
system has
one solution,
solution, which
the
If the
has one
which of the
following can
can NOT
be the
the value
value of aa ??
following
N O T be
B ) ( -‐ 11,, 33))
B)
C ) ((1,
1 ,-‐ 77))
C)
(3, - 17)
D) (3,‐17)
A
A)) -‐ 1
3
B) 3
B)
-
E
4
C) 1
D
D) ) 3
87
CHAPTER 10
EQUATIONS
CHAPTER
10 SYSTEMS
SYSTEMS OF
OFEQUATIONS
3x
= 15
3x -‐ 63)
6y =
15
= -8
4x‐‐y=‐8
3
1
4x - - y
‐-2x
2x + 4
4yy=
= -‐ 1100
x ++116
6
y ==44x
How
solutions (x,y)
are there
H
o w many
many solutions
(x,y) are
there to the
the
system of equations
equations above?
above?
system
What
the solution
the system
What is
is the
solution (x,y)
(x,y) to
to the
system of
equations
above?
equations above?
A) Zero
Zero
A)
A)
A ) ((-‐ 22,8)
8)
8))( (- 1,12
1,12))
C)) , (1,20)
( 120)
D)) ( (3,28)
3 , 28)
8)
One
B) One
C)) Tw
Twoo
C
More than
than ttwo
D) More
wo
mxx-‐ 66yy =
= 1100
m
2xx-‐ n
nyy =
2
z5
0 .0.5x
5 x ++114
4
y :=
x -‐ yy ==‐ -118
8
_
X
ln
the system
system of equations
equations above,
above, m
m and
and n are
are
In the
the system
system has
constants. If
constants.
If the
has infinitely
infinitely many.
many
According
above,
According to the
the system
system of equations
equations above,
what
is
the
value
of
y
?
what the value
?
.
.
m
so lutions, what
what is
the value
m ??
solutions,
15the
value of g
11
1
A) 12
fi
A)
1
B)§
8) 3
4
C)
C) 5
3
D
D) ) 3
1l
1
3x
y ‘ -a6y
l l ‐= 44
6 x-‐ aayy =
=8
6x
:\/E+3
y=
vx + 3
ln
equations above,
[n the
the system
system of equations
above, aa is aa constant.
constant.
If the
the system
the value
system ha
hass no
no solution,
solution, what
what is the
value
of aa??
A)
m_
fu -
y =3
(x, y) is the
the solution
solution to the
the system
system of equations
equations
IfIf (x,
above,
the value
value of y ??
above, what
what is the
1
3
8)
B) 1
C) 3
D) 6
88
PANDA
THE
COLLEGE PANDA
THE COLLEGE
allowed on the following
calculator is allowed
A calculator
following
questions.
questions.
B
0
@
medium,
small, medium,
sells jelly in small,
supermarket sells
A local
local supermarket
much
as
weigh
Sixteen small
and
large jars.
jars. Sixteen
small jars
jars weigh as much
and large
Four small
large jar. Four
jars and
medium jars
as
w o medium
and one
one large
small
as ttwo
weight
same weight
the same
have the
medium jar have
one medium
jars
and one
jars and
the
small jars
many small
How many
large jar. How
as
jars have
have the
one large
as one
large jar?
one large
weight
weight of one
A)
7
A) 7
B) 8
depending on
points depending
A game
on
rewards points
darts rewards
game of darts
and
regions, A and
are ttwo
There are
which
w o regions,
hit. There
region is hit.
which region
hitting
darts, hitting
throws 3 darts,
above . James
shown above.
B,as
James throws
B, as shown
total of
a total
twice, for a
region B twice,
and region
once and
region A once
region
hits
but hits
darts, but
throws 33 darts,
18points.
also throws
Oleg also
points. Oleg
18
regions
once for a
total of
a total
region B once
and region
twice and
A twice
regions A
rewarded for
are rewarded
many points
How many
points. How
21 points.
21
points are
once?
hitting
region B once?
hitting region
B) 8
9
C) 9
D) 10
10
are
points are
30 questions, 5 points
with 30questions,
test with
math test
a math
On
On a
points
correct answer
rewarded
answer and
and 2 points
each correct
rewarded for each
answer . If James
incorrect answer.
each incorrect
are
James
deducted for each
are deducted
points,
scored 59
and scored
questions and
answered
59 points,
the questions
answered all the
systems of
following systems
the following
which of the
solving
solving which
answers,
correct answers,
number of correct
his number
equations
gives his
equations gives
the
answers,
incorrect
of
x, and
incorrect answers, y, on
on the
number
his number
and his
rectangular
tables, rectangular
A restaurant
types of tables,
two types
has two
restaurant has
circular
ones
people and
and circular
seat 4 people
each seat
can each
that can
ones that
people
If 144 people
tables
people. It
seat 8 people.
each seat
can each
that can
tables that
restaurant,
the restaurant,
at the
tables at
aU 30 tables
are
enough to fill all
are enough
restaurant
the restaurant
does the
tables does
how
rectangular tables
many rectangular
how many
have?
have?
math
test?
math test?
A)
B)
B)
59
xX +
+y
y =
= 59
5x -‐ 2y
Zy =
= 30
30
5x
x
X
+
= 30
+ y!f =
A) 12
A) 12
5x
= 59
59
2y =
5x + 2};
c)
C)
B) 16
16
B)
C)
C)
D)
D)
x+
+y =
= 30
2x
Sy = 59
2x ‐- 53,
D
D)) xX++yy ==3300
5 x-‐ 22yy =: 559
9
5x
89
20
20
24
24
CHAPTER 10 SYSTEMS OF EQUATIONS
In the
7x +
+7
the xy-plane,
xy-plane, the
the graph
graph of y =
= x2
x 2 ‐- 7x
7
intersects the
the graph
intersects
graph of y =
: 2x
the points
2x -‐ 1 at the
points
and (p,q).
(p, q). What
(1, 1)
1) and
What is
is the
the value
value of p
p??
y
_
xx22- ‐2
2x x==yy-‐ 1
1
x = y ‐- 111
1
A system
system of ttwo
equations is graphed
graphed in the
the
w o equations
xy-plane
above.
Which
of
the
following
is
the
xy-plane above. Which the following is the
solution
(x,y)
the system?
system?
solution (x,
y) to the
If (x,
(x, y) is a
a solution
solution to the
the system
system of equations
If
equations
above, what
what is one
one possible
possible value
above,
value of y ??
(0, - 6)
A) (0,‐6)
A)
8)) (B
( ‐ 33,,-‐ 33))
C) ( - ~, - 3)
c>
(4-3)
D) (-3,
-;)
r»
(+2)
x2 _
x2
-
1
y2=_ -11_2
yz
12
xX -‐ 2y
: 00
2y =
lf the
the ordered
Y1) and
satisfy
If
ordered pairs
pairs (x1,
(x1,y1)
and (x2, yi)
y2) satisfy
the system
system of equations
equations above,
above, what
what are
are the
the
the
values of y1 and
values
and 1/2
y2 ?
?
1
and 2
1
1
B
‐ ‐ ‐ a and
n d -- ‐ ‐
B) ---
A)
--
1
2
\/12
v'12
)
12
v'12
1
1
-- and
and E
C)
C) ‐i
4
4
D)
D)
--
1
6
1
and and1
6
90
ll
Inequalities
Inequalities
Just
had equations
equations and
and systems
systems of
of equations,
equations, we
we can
and systems
systems of
of inequalities
inequalities..
Just aswe
as we had
can have
have inequalities
inequalities and
The
you m
u s t reverse
reverse the
the sign
sign every
time you
multiply or
or divide
sides by
The only
only difference
difference is that
that you
must
every time
you either
either multiply
divide both
both sides
by aa
negative
negative number.
number.
For
For example,
example,
2x +
+ 33 < 99
2x
reverse the
point? Well, we
we would
subtract by
by 33 to
to get
get 2x
and then
by
Do we
we have
have to reverse
the sign
sign at
at any
any point?
would subtract
2x < 6
6 and
then divide
divide by
2 to get
get x < 3. Yes, we
we did
did aa subtraction
subtraction but
but at
at no
no point
point d
i d we
we multiply
multiply or
or divide
by aa negative
negative number.
did
divide by
number.
Therefore, the
the sign
stays the
the same.
same.
Therefore,
sign stays
Let’s take
Let's
take another
another example:
example:
3x ++ 55 < 4x + 4
3x
The first step
step is to combine
combine like
like terms.
terms. We
Wesubtract
both sides
by 4x
to get
x’s on
the left
left hand
We then
then
The
subtract both
sides by
4x to
get the
the x's
on the
hand side.
side . We
subtract
subtract both
both sides
sides by 5 to get
get the
constants on the
the right
right hand
the constants
hand side:
side :
3 x-‐ 4
3x
4xx << 4 -7 5
‐ x << ‐- ’ 1l
-x
Notice
Notice that
that the
the sign
hasn’t changed
changed yet.
yet. Now,
N o w, to get
get rid
rid of the
negative in
x, we
multiply
sign hasn't
the negative
in front
front of
of the
the x,
we need
need to
to multiply
both
sides by -‐ 1.
1. Doing
so
means
we
need
to
reverse
the
sign.
both sides
Doing so means
need
reverse the sign.
x>
> ll
This concept
concept is the
so many
many silly
silly mistakes
mistakes that
that it’s
important to
reiterate it.
working with
with negative
This
the cause
cause of so
it's important
to reiterate
it. Just
Just working
negative
numbers
does NOT
N O T mean
m e a n you
you need
change the
sign. Some
negative
numbers does
need to change
the sign.
Some students
students see
see that
that they’re
they're dividing
dividing aa negative
number
reverse the
sign. Don't
reverse the
the sign
sign when
when you
you multiply
or divide
divide
number and
and impulsively
impulsively reverse
the sign.
Don't do
do that.
that. Only
Only reverse
multiply or
both sides
negative number.
number.
both
sides by a
a negative
91
91
CHAPTER 11
CHAPTER
11 INEQUALITIES
INEQUALITIES
EXAMPLE1: Which of
the following
EXAMPLE
ofthe
following integers
solution to the
inequality -3x
‐ 3 x -‐ 7 $
5 -‐ 77xx -‐ 27
27?
integer:s is a
a solution
the inequality
?
A ) -‐ 6
A)
B)
B )-‐ 3
0)4
D
)4
C
)1
C)J
-‐ 33xx -‐ 7 §s- ‐ 77x
x ‐- 227
7
41'5‐20
4x
s - 20
x S
g ‐- S
X
5
did we
we multiply
or divide
divide by
by aa negative
negative number
number so
need to
to reverse
the sign.
sign. We
At no
no point
point did
multiply or
so there
there was
was no
no need
reverse the
We
divided
divided a
0 , but
did soby
positive
a negative
negative number,
number, -‐ 220,
but we
we did
so by aa positi
ve number,
number, 4.
§J .
The
only answer
The only
satisfies x3:S
answer (A) .
answer choice
choice that
that satisfies
S -‐55 is -‐ 66,, answer
EXAMPLE 2: If --77 $
2x +
$ 15,
EXAMPLE
5 -‐-2x
+ 33 5
15, which
the following
following must
be true?
true?
which of the
must be
A
A)) 55$5 xx $$ 6
B ) -‐ 66_$<x_ $
x 5-‐ 5
B)
C)‐6§x$5
D )-5‐ 5$5 xx $5 6
D)
So how
how do
do we
we solve
So
solve these
these "two-inequalities-in-one"
”two‐inequalities-in-one” problems?
problems? Well, we
into two
t w o inequalities
inequalities
we can
can split
split them
them up
up into
that we
can solve
that
we can
solve separately:
separately:
-‐77 5
s ‐2x
- 2x ++ 3
3
-‐2x
2x ++ 33 5
s 15
15
Solving the first inequality,
Solving
inequality,
-‐77 3
$ ‐2x
- 2x ++ 3
3
§ ‐- 22xx
-‐ 1100 s
5 ~2 xx
S
Solving
Solving the second
second inequality,
inequality ,
5
-‐ 22xx ++33§S 115
2
-‐ 22xx $§ 112
x 2 ‐- 66
X ~
Putting
the two
t w o results
get -‐66 S xx S
5 5.
5. Answer
Answer ~-(C) .
Putting the
result s together,
together, we
we get
EXAMPLE 3: To follow
follow his diet
plan, James
James must
must limit
limit his
most 40
40 grams.
grams.
EXAMPLE
diet plan,
his daily
daily sugar
sugar consumption
consumption to
to at
at most
One
has 5 grams
sugar and
and one
fruit salad
contains 7 grams
sugar. If
If James
James ate
ate only
cookies
One cookie
cookie has
grams of sugar
one fruit
salad contains
grams of sugar.
only cookies
and
fruit salads,
following inequalities
inequalities represents
represents the
possible number
number of
cookies c
fruit
and fruit
salads, which
which of the
the following
the possible
of cookies
c and
and fruit
salads ss that
he could
eat in one
within his
his diet’s
sugar limit?
limit?
salads
that he
could eat
one day
day and
and remain
remain within
diet's sugar
5 7
A)§+§<40
A) -+-<
40
C
S
5 7
B)§+§§40
B)
- + - ::;40
C
S
C ) Sc
S c++ 7s
7 s< 4400
C)
D
5 c+
+7
$ 4400
D)) Sc
7ss ::;
The total
a m o u n t of
sugar he
gets from cookies
is Sc.
5c. The
The total
75.
The
total amount
of sugar
he gets
cookies is
total amount
amount of
of sugar
sugar he
he gets
gets from
from fruit
fruit salads
salads is
is 7s.
50
his
total
sugar
intake
for
any
given
day
is
5c
+
75,
and
since
it
can’t
be
m
o
r
e
than
40
grams,
Sc+
75
S
40.
So his total sugar intake
any given day is Sc + 7s, and since it can't be more than 40 grams, Sc + 7s s 40.
Answer
Answer ~(D) .
92
PANDA
COLLEGE PANDA
THE COLLEGE
THE
1?
- x -‐ 1?
mean for y > ~x
does it mean
What does
inequality look like? What
an inequality
does an
what does
standpoint, what
graphing standpoint,
a graphing
From a
From
y
X
----
represents all the points
inequality y > -‐xx -‐ 1 represents
above, the inequality
region above,
shaded region
As shown
points above
above the line
shown by the shaded
look at
below, just
what's below,
and what’s
line and
a line
above a
what's above
track of what’s
keeping track
time keeping
hard time
a hard
have a
y=
just look
at the
you have
If you
- x ‐- 1. If
= ‐x
region. The
"above" region.
the ”above"
always in the
y-axis is always
part of the y‐axis
two parts
into two
cuts the y-axis into
y-axis. The line cuts
parts.. The top part
with
intersection with
show the intersection
doesn't show
graph doesn’t
"below" region.
always in the ”below"
part of the y-axis is always
bottom part
bottom
region. If the graph
and
"above" and
the ”above”
determine the
graph to determine
the graph
through the
vertical line through
draw your
always just draw
the y-axis, you
your oown
w n vertical
you can always
regions.
"below" regions.
“below”
itself do
line itself
the line
points on the
- x -‐ 1, the points
and NOT y z= ‐x
Because y > -‐xx -‐ 1 and
dashed . Because
line is dashed.
that the line
note that
Also note
line
on
points
and
solid,
be
would
line
the
then
1,
x
2::
y
were
inequality . If the equation
the inequality.
satisfy the
not
n
o t satisfy
equation were 2 ‐x ‐ then
line would be solid, and points on the line
inequality .
the inequality.
satisfy the
would satisfy
would
y
- X
example,
inequalities? For example,
system of inequalities?
a system
about a
what about
But what
y ::;
5 ‐x
- x ++ 44
1
-x- 3
yY >
- ~2 x ‐ 3
inequalities. In this
both inequalities.
satisfy both
that satisfy
points that
with the points
region with
goal is to find the region
comes to graphing,
When it comes
When
graphing, the goal
1
.
.
can
we can
points, we
this set of pomts,
locate this
= ~x
y=
above 31
but above
below y = -‐xx +
that are below
points that
want the points
we want
case, we
+ 4 but
Ex ‐- 3.
3. To locate
1x-
-x
1
.
overlap .
the rregions
where the
and see where
and above
= ‐x ++ 4 and
below y z
regions below
shade the regions
shade
above y = Ex -‐ 3 and
e g i o n s overlap.
y
X
system.
solutions to the system.
that are solutions
points that
the points
contains all the
the left contains
The overlapping
on the
region on
overlapping region
93
93
CHAPTER 11
CHAPTER
11 INEQUALITIES
INEQUALITIES
Now
N o w if
system as
asifif it were
system of equations
equations instead
system of
of inequalities,
we would
would
if we
we solved
solved the system
were a
a system
instead of a
a system
inequalities, we
the intersection
intersection point
point of the
get the
the ttwo
w o lines,
case, happens
be the solution
solution with
with the
highest value
value
lines, which,
which, in this case,
happens to be
the highest
of x. As an
find this
this solution.
Substituting the
the first "equation"
”equation” into
second, we
get
an exercise,
exercise, let’s
let's find
solution. Substituting
into the
the second,
we get
11
‐ x ‐- 3
-‐ xx+
+ 4 == -2x
2
:x‐6
-‐ 2
2xx++88=x4
-‐ 33xx = ‐- 114
.Xr
14
‐ z ;:::::
44.66
.66
= z‐__3
3
- 14
this from
from the
the first equation).
Therefore, (4.66,
is the
the solution
solution
At x
x = 4.66, y = -‐4.66
4.66 +
+ 4 = -‐0.66
0.66 (we
(we get
get this
equation). Therefore,
(4.66, -‐O.66)
0.66) is
with the
the highest
with
There are
no
solutions
in
which
x
is
5,
6,
or
larger.
highest value
value of x. There
are no solutions
which
larger.
While finding
finding the
While
the intersection
example may
may have
(hahal), these
these points
intersection point
point in this
this example
have seemed
seemed a bit
bit pointless
pointless (haha!),
points can
can
be very
be
very important
important in the context
context of a given
given situation,
situation, such
such as
maximize profit
as finding
finding the
the right
right price
price to maximize
profit or
figuring
right amount
materials for a construction
construction project.
figuring out
out the
the right
amount of materials
project.
EXAMPLE
EXAMPLE4:
4:
y
II
I
lII
IV
The following
following system
system of inequalities
inequalities is graphed
graphed in the xy-plane above.
2::‐- 33xx + 1
y2
y 22::22xx -‐ 3
Which quadrants
quadrants contain
Which
solutions to the
the system?
contain solutions
A) Quadrants
Quadrants I and
and JI
II
Quadrants I and
and IV
B) Quadrants
C) Quadrants
Quadrants III
I I I and
C)
and IV
D) Quadrants
I, II,
I I , and
D)
Quadrants I,
and IV
IV
First, graph
equations, preferably
your graphing
graphing calculator.
shade the
the
First,
graph the
the equations,
preferably with
with your
calculator. Then
Then shade
the regions
regions and
and find the
overlapping
region.
overlapping region .
.1/
X
1
As you
see, the
overlapping region,
which contains
contains all
region.. It
has points
in
As
you can see,
the overlapping
region, which
all the
the solutions,
solutions, is
is the
the top
top region
It has
points in
quadrants I, II,
I I , and
quadrants
and IV. Answer
Answer ~.(D) .
94
THE
PANDA
THE COLLEGE
COLLEGE PANDA
Ecologists have
have determined
determined that
the number
number of
EXAMPLE 5: Ecologists
that the
of frogs
frogs y
y must
must be
be greater
greater than
than or
or equal
equal to
to tluee
three
times
ecosystem to
in a
forest. In
In addition,
times the number
number of snakes
snakes x for a
a healthy
healthy ecosystem
to be
be maintained
maintained in
a particular
particular forest.
addition,
the
number of
the number
frogs and
number of
of frogs
and the
the number
of snakes
snakes must
must sum
sum to atwleast
at.least 400.
PART
Which of the
systems of
of inequalities
inequalities expresses
conditions for
for a
healthy ecosystem?
ecosystem?
PART1:
l! Which
the following
following systems
expresses these
these conditions
a healthy
y 2~33xx
y ‐- xx>400
>400
A)
A)
y 2~ 33xx
y‐x2400
y 2~33xx
y+12400
B)
B)
C)
C)
D)
D)
y ~ 3x
3,53:
y+x5400
PART2: IfIf the
the forest
forest currently
currently has
PART
has a healthy
healthy ecosystem,
ecosystem, what
what is the
the minimum
number of frogs
in
minimum possible
possible number
frogs in
the forest?
the
forest?
Part 1 Solution:
Solution: The number
number of frogs, y, must
must be
be at
at least
least three
three times the number
number of snakes,
snakes, x.
x. So,
50, y 2".
2 3x.
3x. The
The
number of frogs
frogs and
and the number
number of snakes
snakes must
soyy +
number
must sum
sum to at least
least 400, so
+ x 2".
2 400. Answer
Answer ~-(C) .
these types
types of questions,
the strategy
the minimum
minimum in the
the graph
graph of the
Part 2 Solution:
Solution: In
ln these
questions, the
strategy is to look for the
the
inequalities. The minimum
minimum (or maximum)
maximum) will
typically occur
occur at
at the intersection
intersection point.
point. To show
show you
you what
what l1
inequalities.
will typically
mean, let's
put the
second inequality
inequality in y =
mx +
mean,
let’s first put
the second
= mx
+ b form.
33xx
y 22".
00
y 22:‐ -x x++4 400
Now
graph the
the inequalities
inequalities using
using aa calculator
Now we
we can graph
calculator..
y
The graph
graph confirms
confirms that
that y,
y, the
the number
number of
at a minimum
minimum at the intersection
point. After
all, the
the
The
of frogs,
frogs, is
is at
intersection point.
After all,
overlapping
region
(the
top
region)
represents
solutions and
and the
the intersection
intersection point
point is at the
the bottom
bottom
overlapping region (the top region) represents all
all possible
possible solutions
of this region,
region, representing
representing the solution
solution with
minimum number
number of frogs.
of
with the minimum
We
can find
find the
the coordinates
of that
that intersection
solving a
a system
system of equations
We can
coordinates of
intersection point
point by solving
equations based
based on
on the ttwo
wo
lines.
lines.
y=
= 3x
3x
= -‐xx + 400
y=
Substituting the
the first equation
equation into
into the second,
second,
Substituting
= -‐xx +
+ 400
3x =
: 400
4x =
x = 100
X
So, 100
100 is
is the
the x-coordinate.
x-coordinate. The
Theyy‐coordinate
-coordinate must
beyy := 3x
3x =
Given these
these values,
values, the
So,
must then
then be
= 3(100)
3(100) =
: 300. Given
the
I
intersection point
point is
is at
at (100,300
and the
minimum possible
possible number
number of frogs is j 300 when
has a
intersection
(100,300)) and
the minimum
when the forest has
health y ecosystem.
healthy
ecosystem.
95
95
CHAPTER 11
CHAPTER
11 INEQUALITIES
INEQUALIT'IES
page 299.
chapter start
this chapter
EXERCISE:Answers
CHAPTER EXERCISE:
Answers for this
start on page
following
allowed on the following
calculator is allowed
A calculator
questions.
questions.
y
the
to the
solution to
a solution
following is a
the following
Which of the
Which
14?
4x ‐- 14?
4 > 43:
- x -‐ 4
inequality ‐x
inequality
------.F------+
A)) -‐ 1
A
X
B) 2
C) 5
D) 8
Which of the following systems of inequalities
could be the one graphed in the xy-plane above?
A) y > 3
A
)y>3
1
IfIf 3Zx‐4>
~x - 10, which of the following
~ x - 4 > Ex‐10,whichofthefollowing
yy >
> xx
must be true?
mustbetrue?
B) y < 3
B) y < 3
A
4
A)) x <<224
B
4
B)) x >>224
y <x
y<x
C
< 33
y<
C)) y
yy >
> xx
< ‐-224
C
4
C) ) x <
D
4
> ‐-224
D)) x >
D
D)) yy >> 3
y<
<x
marbles in a jar.
Jerry
are m marbles
there are
that there
estimates that
Jerry estimates
number of marbles
Harry,
marbles
actual number
knows the actual
who knows
Harry , who
number, n, is
in the
actual number,
the actual
that the
notes that
the jar, notes
estin1ate .
Jerry's estimate.
within
(inclusive) of Jerry's
marbles (inclusive)
within 10 marbles
represents
Which
inequalities represents
following inequalities
the following
Which of the
the
the
and the
estimate and
Jerry's estimate
between Jerry’s
relationship between
the relationship
the
actual
number
of
marbles
in
the
jar?
marbles
number
actual
II ‐- 110
A)) 11
A
n+
+ 10
1 0'.S
5 mm '.S
5n
0
B
m-‐ l10
O '.S
S nn S'.S
nm
H+
- I10
O
B)) m
C
n '.S
Sm
l 1011
On
m S'.S
C)) 11
m
< n < 10m
D)
D) ‐-10_n_10m
10 <
96
R
E
THE
THE COLLEGE
COLLEGE PANDA
PANDA
‑
manufacturer produces
A manufacturer
produces chairs
chairs for a
a retail
retail store
store
according to the
according
formula,
M
=
12P
+
100,
where
the formula, M = 12P +
where
M is the
P is
the number
number of units
units produced
produced and
and P
is the
the
retail price
price of each
retail
units
each chair.
chair. The number
number of units
sold by the
the retail
sold
store is given
retail store
given by
= ‐- 33PP + 970, where
N=
where N is the
the number
number of
of units
units
sold and
and P is the retail
sold
What
retail price
pri ce of each
each chair.
chair. What
are all the
the values
values of P for which
are
which the
the number
number of
of
units produced
produced is greater
units
to the
greater than
than or
or equal
equal to
the
number of units
number
sold?
unit s sold?
‘
yy
f'--------1--------.i
A) P
P2
~ 58
58
A)
B) P
P5
B)
:S 58
58
X
The
The graph
graph in the
the xy-plane
xy-plane above
above could
could represent
represent
which
which of
of the
the following
following systems
systems of
of inequalities?
inequalities?
C) P
55
P 2:::::
55
A)
3
A) y
y 2:::::
3
:s
D) P g
:S55
y 5 ‐3
-3
B) y
yS
:S 33
B)
y
:::::
y z ‐- s3
If n is an
an integer
integer and
and 3(n ‐- 2) > ‐4(n
If
- 4(n -‐ 9),
9), what
what
is the
the least
least possible
possible value
value of n
n?
?
C)
2 3
C) x
X:::::
3
r < _3
X
'.S- 3
D) x S
:S 3
xX 22 -‐33
work, Harry
To get to work,
Harry must
must travel
travel 8 miles
miles by bus
bus
and 16
miles by train
train everyday.
and
16miles
everyday . The
The bus
bus travels
travels
an average
average speed
speed of x miles
at an
miles per
per hour
hour and
and the
the
train travels
travels at
train
speed of y miles
miles per
at an
an average
average speed
per
hour . If
lf Harry’s
Harry's daily
hour.
daily commute
commute never
never takes
takes more
more
than 1 hour,
hour, which
which of the
than
the following
following inequalities
inequalities
represents the
the possible
represents
possible average
average speeds
speeds of the
the bus
bus
and train
train during
during the
and
the commute?
commute?
8 16
A)§+ES1
A) -+-:S
l
X
y
x
16
y
8
x
y
B) -+-:S
B>E+§g1
l
X
y
yy
- +-‐<
C) _
<1
C)8+16‐1
8 16 xX
D
8 x + 16y
1 6 y:S
§ 11
D)) Bx+
97
CHAPTER
CHAPTER 11
11 INEQUALITIES
INEQUALITIES
An
out to
An ice
ice cream
cream distributor
distributor contracts
contractsout
to two
two
different companies
different
companies to
to manufacture
manufacture cartons
cartons of
produce 80
cartons
ice cream
ice
cream.. Company
Company A can
can produce
80 cartons
each hour
each
hour and
and Company
Company B can
can produce
produce 140
cartons
The distributor
distributor needs
fulfill
cartons each
each hour.
hour. The
needs to
to fulfill
an order
order of over
an
over 1,100
1,100 cartons
cartons in 10
10 hours
hours of
contract
contract time.
time. It contracts
contracts out
out x hours
hours to
remaining hours
hours to
Company
Company A and
and the
the remaining
inequalitie s
Company
Company B. Which
Which of the
the following
following inequalities
gives
possible values
gives all
all possible
values of xx in the
the context
context of
this problem?
problem?
this
80
3
y2§x+2
Y -> -2 x + 2
5
y g::;‐-Z2xx -‐ S
Which of the
the following
following graphs
the xy-plane
xy-plane
Which
graphs in the
could represent
represent the
the system
system of inequalities
inequalities above?
above?
could
A)
A)
140
~>
1100
>1,100
X +
A) ‘x‐
+ 10
10‐- Xx
A)
I
1, 100
B) 140x +
+ 80(10 -‐ x) >
>1,100
140(10 -‐ x) > 1,100
C) 80x
80x +
+140(10
1,100
B)
B)
100
D) 80x +
+ 140(x
140(x -‐ 10)
10) > 1,
1,100
5 xx +
y >>1l5
+aa
C)
C)
y<
<55xx +
+bb
inequalities above,
above, a and
and bare
In the
the system
system of inequalities
b are
constants . If
lf (1,
(1,20
constants.
20)} is a solution
solution to the
the system,
system,
which
the following
could be
which of the
following could
be the
the value
value of
of
b -‐ aa ?
D)
D)
A)
A) 6
B) 8
C ) 110
0
C)
D ) 112
2
D)
98
THE COLLEGE
THE
COLLEGE PANDA
PANDA
no more
more than
than 30
Tina works
works no
30 hours
hours at
at a
a nail
nail salon
salon
each week.
week. She
She can
can do
do a
each
a manicure
manicure in 20
20 minutes
minutes
and a
a pedicure
pedicure in 30
and
manicure
30 minutes.
minutes . Each
Each manicure
earns
her $25 and
and each
pedicure earns
earns her
earns her
each pedicure
her $40,
and
must earn
and she
she must
earn at least
least $900 to cover
cover her
her
expenses. If
expenses.
week, she
she does
does enough
enough
If during
during one
one week,
manicures m and
manicures
and pedicures
pedicures p to cover
cover her
her
expenses,
of
expenses, which
which of the
the following
following systems
systems of
inequalities describes
inequalities
describes her
her working
working hours
hours and
and her
her
earnings?
earnings?
A)
A)
B)
If -
9
.
< - 2x + 4 < - , what 1s one possible
2
value of x - 2 ?
+ 2p ::;
5 30
3m +
25m + 40];
40p 2 900
Joyce
create a
that
Joyce wants
wants to create
a rectangular
rectangular garden
garden that
has
least 300 square
has an
an area
area of at least
square meters
meters and
and a
a
perimeter
perimeter of at
at least
least 70
70 meters.
meters . If the
the length
length of
the
and the
is yy
the garden
garden is x meters
meters long
long and
the width
width is
meters long,
long, which
meters
of the
following systems
systems of
which of
the following
of
inequalities
inequalities represents
represents Joyce’s
Joyce's requirements?
requirements?
2m +
+ 3p S
::; 30
25m
25m +40p
+ 40p 2 900
C)
C)
20
3
p
flm
-+-<
30
3 +2E<
_30
3 225m+40p2900
25m
+ 40p 2 900
A)
xy 2 70
x+
+ y 2 300
300
D)
D)
_"1
E>
Ill
p 900
3-+-2_900
-+->
3 2-
B
y 221 150
50
B)) x xy
xX + y.1/22770
0
25m + 40p ::; 30
25m+40p§30
C
y 223 300
00
C) ) x xy
xX + y.1/22770
0
If k 5
3k+
which of
of the
the following
following must
::; x 5
::; 3k
+ 12, which
must
be true?
be
true?
I.
lII.
l.
III.
111.
D
y 223 300
00
D) ) x xy
xX +
+ y.1/223355
x ‐- 12
12 5
::; 3k
k 2 -‐ 66
20
x -‐ k 2
IfIf a
a < b, which
which of
of the
the following
following must
must be
be true?
true?
A) lonly
I only
B ) Il aand
n d lIIl only
B)
only
C) llII and
C)
I I only
and IlII
only
D) I,I, 11,
D)
II, and
and 111
III
a2 <
< b2
b2
I. a2
1],
2a <
< 2b
II. 2a
I111.
I ] . -‐ b <
< -‐ a
A) 11
II only
only
B
n d lIIl only
B)) lIaand
only
C) II and
[
I
and III] only
only
D
I , and
ll
D)) II,, III,
and IIII
99
12
Problems
WordProblems
Word
to translate
you to
They require
experience. They
a frustrating
problems is
word problems
solving word
For
students, solving
is a
frustrating experience.
require you
translate the
the
many students,
For many
show
will
chapter
this
in
exercises
the
and
math. The
even do the
can even
before you
question
you can
the math.
The examples
examples and the exercises
chapter will show you
you how
how
question before
words
translating words
instinct for
an instinct
develop an
to handle
handle the
the full range
range of word
word problems
problems that
that are
are tested.
tested. You will
will develop
for translating
guide.
best guide.
the best
Experience is the
and finally
variables, and
right variables,
the right
setting the
into math,
finally solving
solving for the answer.
answer. Experience
math, setting
into
of these
the largest
What is the
EXAMPLE
EXAMPLE 1: The
The sum
sum of three
three consecutive
consecutive integers
integers is 72. What
largest of
these three
three .integers?
integers?
things
the things
of the
one of
be one
variable be
a variable
problems is to let a
important technique
most important
The
The most
technique in solving
solving word
word problems
be x.
one
smallest
the
let
we
so
integers,
three
this problem,
know. In this
know.
problem, we don't
don’t know
know any
any of the
the three integers, so
let the smallest one be
x.
problem.
the
long as
as long
as x, as
m a t t e r which
which number
number we set as
as we're
we’re consistent
consistent throughout
throughout the problem.
matter
don't
you don’t
you
doesn't
It
It doesn’t
smallest, then
the smallest,
So
if xis
x is the
then our
o u r consecutive
consecutive integers
integers are
are
So if
x, x +
+ 1,x
1,x +
+ 2
x,x
can make
Because they
Because
they sum
sum to 72, we
we can
make an
an equation:
equation:
+ ((xx ++ 11)) + ((xx + 22)) =: 7722
xx +
3 x+
+ 33 z=772
2
3x
3 x=
: 669
9
3x
xX :=223
3
largest) .
(the largest).
and I25
be 23, 24, and
then be
three consecutive
smallest, our
the smallest,
Because
x is the
o u r three
consecutive integers
integers must
must then
25 I(the
Because xis
three integers
Our
integer? O
largest integer?
be the largest
But what
what would
would the
the solution
solution have
have looked
looked like
like if we
we had
had let
let x be
u r three
integers would've
would’ve
been
been
2,.\‘ -~ 1,x
1, x
x -‐ 2,x
been
would've been
A n d our
o u r equation
equation would’ve
And
(x - 2) + (x - 1) + x = 72
3 x-‐ -3 =
‐ 772
2
3x
3
‐ 775
5
3xr =
rX ‐=225
5
answer!
the answer!
already at the
we're already
scenario, we’re
this scenario,
A n d because
was set
set to be the
largest of the
the three
three integers
integers in this
the largest
because x was
And
100
THE COLLEGE
PANDA
THE
COLLEGE PANDA
The lesson
lesson here
which unknown
variable. Often
times, that
that
here is that
that you
you should
should think
think about
about which
unknown you
you want
want to
to set
set as
as the
the variable.
Often times,
unknown
be what
is asking
asking for.
for. Other
Other times,
to
unknown will
will be
what the question
question is
times, itit will
will bean
be an unknown
unknown you
you specifically
specifically choose
choose to
make
solve. And
And sometimes,
sometimes, as
1, itit doesn’t
make the problem
problem easier
easier to set up and
and solve.
as was
was the
the case
case in
in Example
Example 1,
doesn 't matter
matter
which
with the
same answer
which unknown
unknown you
you pick; you’ll
you'll end
end up
up with
the same
answer with
with the
the same
same amount
amount of
of effort.
effort.
EXAMPLE 2:
2: One
One number
number is
is 3 firms
another number.
number. U
If they
EXAMPLE
times another
they sum
sum to
to 44,
44, what
what is
is the
the larger
larger of
of the
the two
two
numbers?
numbers?
In
want to
be the
the smaller
smaller of
1n this problem,
problem, we
we want
to set xx to
to be
of the
the two
two numbers.
numbers. That
That way,
way, the
the two
two numbers
numbers can
can be
be
expressed as
expressed
as
xx and
and 3x
3x
If we
we would
have to
to work
we let xx be
be the larger
larger of
of the
the two,
two, we
would have
work with
with
xX
x.r an
andd ‑3
3
and fractions
and
fractions are
are yucky.
Setting
u r equation,
Setting up oour
equation,
xX + 3x
3x =
: 44
44
4x
4x =
= 44
44
xX =
= 11
Becareful‐we’re
asks for
for the
Be careful - we're not
not done
done yet! The
The question
question asks
the larger
larger of
of the
the two,
two, sowe
so we have
have to
to multiply
multiply x
x by
by 3
3 to
to
get I-.33 I.
EXAMPLE
What is
number such
that the
square of
is equal
its reciprocal?
reciprocal?
EXAMPLE 3: What
is a
a number
such that
the square
of the
the number
number is
equal to
to 2.7%
2.7% of
of its
Let the number
number we’re
we're looking
looking for bex.
be x.
1
2
x2
= .027 x1
X =
X -
X
Multiply
Multiply both
both sides
sides by
by x
x to
to isolate
isolate it.
it.
xx33 =
= .027
Cube
both sides.
Cube root
root both
sides .
x:‑
= []]
X
101
PROBLEMS
WORD PROBLEMS
CHAPTER
12 WORD
CHAPTER 12
old
How
Henry. H
as Henry.
old as
as old
twice as
be twice
will be
Albert will
than Henry.
older than
years older
Albert is 7 years
EXAMPLE4:-Albert
EXAMPLE4:
Henry. In 55 years,
years, Albert
o w old
now?
is Albert now?
the
earlier, assigning
as we mentioned
but aswe
age, but
Henry's age,
be Henry’s
assigned x to be
We could've assigned
age nnow.
Albert's age
be Albert’s
Let x be
o w. Wecould’ve
mentioned earlier,
assigning the
might
you
of
some
point,
this
at
Now
route.
faster
the faster route. Now at this point, some you might
typically the
asking for is typically
question is
the question
what the
be what
is asking
variable
variable to be
add
only add
would only
work, it would
certainly work,
would certainly
that would
While that
variable to Henry's
another variable
assigning another
be
thinking of assigning
Henry’s age.
age. While
be thinking
more.
calls for more.
clearly calls
question clearly
the question
stick to one
solution. Try to stick
the solution.
more
one variable
variable unless
unless the
steps to the
more steps
old .
be x -‐ 7 years
must
Henry m
then Henry
old nnow,
years old
Albert is x years
If Albert
If
o w, then
u s t be
years old.
old .
years old.
will be
and Henry
be x +
will be
Five
o w, Albert
Albert will
+ 5 and
Henry will
be x ‐- 2 years
from nnow,
years from
Five years
+
( x -‐ 22))
+ 55 ==22(x
x + 5 = 22xx -‐ 4
x
X
we
first 10
the first
for the
yards per
UO yards
yards per
60 yards
can run
Jake can
EXAMPLE5: Jake
EXAMPLE5:
mm60
per minute.
minute. Amy
Amy can
can rrun
u n 120
per minute
minute for
10minutes
minutes
how
after how
time,
same
the same time, after
at the
running at
start running
they start
thereafter. If they
yards per
20 yards
down to 20
slows down
then slows
but
but then
per minute
minute thereafter.
?
assuming t > 10
distance, assuming
same distance,
the same
Amy have
and Amy
Jake and
will
minutes t w
many minutes
many
i l l both
both Jake
have run
r u n the
10 ?
with Amy's
distance rrun
equate Jake's
want to equate
work with.
variable t to work
us aavariable
gives us
already gives
problem already
The problem
The
with. We
Wewant
Jake’s distance
u n with
Amy’s..
Jake’s
distance: 60t
Jake's distance:
20(t -‐ 10)
distance: 120(10)
Amy's
Amy’s distance:
120(10) +
+ 20(t
10)
60t::120(10)4-20(t‐»10)
= 120(10 ) + 20(t - 10)
60t
60t =
: 11,200
, 2 o o+
+ 20t
2 0 ; - 200
M
= 11,000
bm)
40tn =
tf =z 25fi
the same
they will
minutes, they
After I2sIminutes,
After
will have
have rrun
u n the
same distance.
distance.
the scientists.
shared among
be shared
must be
equipment must
research equipment
company, research
a pharmaceutical
EXAMPLE6: At a
EXAMPLE
pharmaceutical company,
among the
scientists.
for
freezer for
one
and one freezer
scientists, and
3 scientists,
every 3
for every
centrifuge for
one centrifuge
scientists, one
every 4 scientists,
microscope for every
one micromope
There is one
scientists
many scientists
company, how
equipment at this company,
52 pieces
there is a total
scientists. If there
every 2
every
2scientists.
total of 52
pieces of research
research equipment
how many
are there?
i,
. X
.
i,
, x
is 5, and
centrifuges IS
number of centrifuges
the number
is 1, the
U1enumber
Then the
number of scientists.
the number
Let x be
be the
scientists. Then
number of microscopes
microscopes IS
and
. x
the
IS ~
5..
freezers is
number of freezers
the number
X
X
x
X
+ - =
_ 552
2
-Z4 ++ -3 +
2
fractions,
the fractions,
both sides
Multiply both
Multiply
sides by 12
12 to get
get rrid
i d of the
3 x+ 4
2 ~• 12
l2
4xx + 66xx z=552
3x
1
3 x=: 6624
24
13x
x:‑
102
THE COLLEGE
COLLEGE PANDA
THE
PANDA
EXAMPLE
A group
cost of renting
a cabin
cabin equally.
equally. If each
pays $130,
EXAMPLE 7:
7:A
group of friends
friends wants
wants to
to split
split the
the cost
renting a
each friend
friend pays
they
they will
have $50 too
too little.
does it cost
cost
they will
will have
have $10 too
too much.
much. If each
each friend
friend pays
pays $U0,
$120, they
will have
little. How
How much
much does
to rent
rent the
the -cabin?
cabin?
have two
t w o unknowns
problem.. We'll
We’ll let
let the number
group be
We have
unknowns in this problem
number of people
people in the group
be n and
and the
the cost of
renting aa cabin
cabin be
be c.
c. From
From the
the information
information given,
given, we
we can come
come up
w o equations
(make sure
renting
up with
with ttwo
equations (make
sure you
you see the
the
reasoning
reasoning behind
behind them):
13011 -‐ 10 =
z cC
13011
+ 50 = cC
120n +
equation, 13011
130n represents
represents the
the total
total amount
amount the group
10 dollars
dollars too much,
In the first equation,
group pays,
pays, but
but because
because that’s
that's 10
much,
need to subtract
subtract 10
10to
arrive at the
the cost
cost of rent,
rent, c. In the second
second equation,
total amount
equation, 120n represents
represents the total
amount
we need
to arrive
the group
group pays,
pays, but
but this time
time it's
it’s 50
50 dollars
dollars too little,
little, so
50 to arrive
at c.
6. Substituting
from
the
so we
we need
need to add
add 50
arrive at
Substituting c from
the
into the second,
the first equation
equation into
second, we
we get
get
: 130n
130" -‐ 10
120n + 50 =
-‐ 1lOn
0 n=: ‐ 660
0
= 6
n=
So
So there
there are
are 6 friends
friends in the group.
group. And
And
c:=
C
13011‐10=
1 3 0· 6- 6- ‐ 10
1 0== 770
13011
- 10 = 130
renting the cabin
cabin is
is I-.770 1The cost of renting
EXAMPLE 8:
8: Of the
the 200 jellybeans
jellybeans in
in aa jar, 70% are
are green
rest are
red. How
How many
many green
jellybeans
EXAMPLE
green and
and the
the rest
are red.
green jellybeans
must be
be removed
removed so
so that 60% of
of the
the remaining
remaining jellybeans
jellybeans are green?
must
green?
The answer
answer is NOT
N O T 20. You can't
can’t just
just take
take 10% of the
the green
jellybeans away
do that,
green jellybeans
away because
because as
as you
you do
that, the total
total
:a
number
jellybeans also
also goes down.
down. We
We first find that
that there
jellybeans.. We
x 200 = 140 green
green jellybeans
We need
need to
number of jellybeans
there are
are % x
remove x of them
so that
that 60% of what's
what’s left is green:
green:
remove
them so
jellybeans left _ 60“/¼
green jellybeans
green
------= 6O°oo
totaljellybeans
total
jellybeans left ‘
140‐x_
66
140
- X
200
- 110
2 0 0- ‐Xx _
0
Cross multiplying,
multiplying,
: 6(200 -‐ x)
10(140 -‐ x) =
1,400
10x := l,1,200
1,
400 -‐ lOx
200 -‐ 6x
200 =
= 4x
=
x _‐_ 50
X
50
green jellybeans
jellybeans need
need to
to be
be removed
removed.. This
This type
type of
with percentages
is very
very common
IsoIgreen
of word
word problem
problem with
percentages is
common in
in
chemistry and
and is typically
known as
asa
”mixture" problem
problem..
chemistry
typically known
a "mixture"
103
CHAPTER 12 WORD
CHAPTER
WORD PROBLEMS
PROBLEMS
Our next
next example
example is the
Our
area / perimeter word
word problem.
problem.
the classic area/perimeter
EXAMPLE 9: A rectangle
EXAMPLE
rectangle has
that is
is 3
rectangle is
is
has a
a width
width that
3 inches
inches shorter
shorter than
than its
its length.
length. It
If the
the area
area of
of the
the rectangle
square inches,
inches, what
108 square
what is the
the perimeter,
perimeter, in inches,
inches, of
of the
the rectangle?
rectangle?
Em
I
If we
we let the
If
be I, then
then the
the width
w is/
is I -‐ 3. Since
Since aa rectangle’s
the length
the length
length be/,
width w
rectangle 's area
area is
is equal
equal to
to the
length times
times the
the width,
width,
we can
up the
we
can set up
the following
following equation:
equation :
: 108
llw
w=
/(/
- 3) = 108
1(1‐
12‐- 3131 ‐ 108 z= o
12
0
( 1-‐ 112)
2 )(/( 1
):o
(/
+ +9)9 =
0
Since the
width is
then /I -‐ 3
3 =z 12
12‐- 3
Finally, the
the length
length of a
a rectangle
rectangle has
has to be
be positive,
positive, /I =: 12. The width
is then
3 =z 9.
9. Finally,
the
perimeter
21+
210 =
: 2(12) +
+ 2(9) =: I-.42 1.
perimeter is 2/
+ 2w
Never forget
forget that
Never
that the perimeter
perimeter of a
many
a rectangle
rectangle is twice
twice the
the length
length plus
plus twice
twice the
the width.
width . I’ve
I've seen
seen too many
students
students just
just add
add the
the length
length and
and the width
width without
without thinking
thinking it through.
through .
EXAMPLE 10: When
EXAMPLE
When Alex
Alex and
Barry work
work separately
separately from
from each
paint a
and Barry
each other,
other, Alex
Alex can
can paint
a house
house in
in 6
6 days,
days,
and Barry
can paint a house
Barry can
house in 12
12days. Assuming
Assuming that
they
each
work
at
a
constant
rate,
how
many
days
that they each work at a constant
days
will
w
i l l it
it take
Alex and
Barry to paint a house if
if they
they work together?
take.Alex
and Barry
together?
This is the
the typical
typical “work‐rate”
"work-rate" problem
This
problem that
involves two
t w o individuals
different rates.
The general
general
that involves
individuals who
who work
work at
at different
rates . The
approach
approach is to use
use the
r t , where
where W
W is
is the
the amount
of
work
done,
r
is
the
overall
rate
at
which
the formula
formula W = rt,
amount of work done, r is the overall rate at which
work
being done,
work is being
done, and
t
is
the
time
spent.
The
key
thing
to
note
is
that
the
overall
rate,
r,
can
be
found
by
and
the time spent. The key thing
note that
overall rate, r, can be found by
summing up
summing
up the
the individual
individual rates.
rates.
11
Since Alex can
days,, his
his rate
rate is 5 of a house
12
can paint
paint a house
house in 6 days
house per
per day.
day. Since Barry
Barry can
can paint
paint a
a house
house in
in 12
1
days, his
his rate
per day.
days,
rate is ‐ of a
a house
house per
112
2
6
.
. 1
_
1 _ 2
1 _ 3 _ 1
Working
+ 12 =‐ 12 + 12 =‐ 12 = 4 of
N o w we
we can
can use
‐ rt,
rt,
Working together,
together, they can
can pamt
pamt 6 +
of a
a house
house per
per day.
day. Now
use W
W=
6 12 12 12 12 4
where W =
= 1 (i.e.
(Le. 1 house)
house) and
z ~,
31, to find
find the
the time
time it will
will take
paint one
house..
where
and r =
take them
them to
to paint
one house
W=
z rrtt
1
1l z= ‐- t
4
4 =: 1t
Therefore,
paint one
one house.
This answer
makes sense
sense because
Therefore, it will take
take Alex and
and Barry
Barry [±]days
days to
to paint
house. Thjs
answer makes
because ifif Alex
Alex
can
days by himself,
then
it
should
take
less
than
6
days
if
Barry
is
working
can finish
finish a
a house
house in 6 days
himself, then it should take less than 6 days if Barry is workjng alongside
alongside
him.
him.
104
THE COLLEGE
COLLEGE PANDA
PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 301.
A calculator should
N O T be
should NOT
be used
used on the
following
following questions.
questions.
A rectangular
rectangular monitor
monitor has
has a
a length
length of x
x inches
inches
and
and a
a width
width that
that is one‐third
one-third of its length.
length. If the
the
perimeter
48 inches,
what is the
the
perimeter of the
the monitor
monitor is 48
inches, what
va lue of x?
value
x?
Which
Which of the
the following
following represents
represents the square
square of
the sum
sum of x and
the
and y,
y, decreased
decreased by the
the product
product of
and y
y??
x and
A) x2
A)
x2+
+y2
y 2 -‐ xy
xy
2y 2 -‐ xy
B) xx2y2
Jr};
C) (x+y)2‐
(x + y) 2 - ((xH+yy))
C)
D) (x + y)2 -‐ xy
D)
Susie
salmon,
weighing x
Susie buys
buys 2 pieces
pieces of sa
lmon, each
each weighing
pounds,
pounds, and
and 11 piece
piece of trout,
trout , weighing
weighing y
pounds,
salmon
pounds, where
where x and
and y are
are integers.
integers . The salmon
cost $3.50 per
per
per pound
pound and
and the
the trout
trout cost
cost $5 per
pound.
the fish was
was $77, which
pound . If the
the total
total cost
cost of the
which
of the following
be the
value
y?
following could
cou ld be
the va
lue of y?
On a
a 100 cm
cm ruler,
On
are drawn
drawn at
at 10, X, and
ruler, lines
lines are
and
The distance
lines at X and
and
98 cm. The
distance between
between the
the lines
times the
the distance
distance between
98 cm is three
three times
between the
lines
is
the
value of X ??
lines at X and
and 10 cm. What
What the value
A)) 4
A
B) 5
C) 6
D) 7
If 5 is added to the square root of x, the result is
9. What is the va lue of x + 2?
A 20% nickel
alloy was
was made
nickel alloy
made by
by combining
combining 22
grams
nickel alloy with
an
grams of a
a 35% nickel
with 6 grams
grams of an
x%
x % nickel
nickel alloy. What
What is the value
value of xx ?
?
A grocery store sells tomatoes in boxes of 4 or 10.
If Melanie buys x boxes of 4 and y boxes of 10,
where x 2: 1 and y 2: 1, for a total of 60
tomatoes , what is one possible value of x ?
105
PROBLEMS
WORD PROBLEMS
CHAPTER
CHAPTER 12 WORD
following
allowed on the following
A calculator is allowed
questions.
questions.
At a Hong Kong learning center,
i
of the
1
tak e
studen ts take
th e students
debate, 2 of the
tak e debate,
studen ts take
students
6
?
If 8 + 5x is
’ twice
' x -‐ 5,, what
h is
' the value off x ?
8 + 5x 15tw1ce x 5 w at 15the value 0 x
- 6
A) ‐6
A
..
,
The
science . The
take scrence.
students take
the students
~1 of the
and §
writing, and
writlng,
is
what is
math, what
take math,
33 studen ts take
math. If 33students
take math.
rest take
rest
learning
the
at
students
of
number
total number students at the learning
the total
the
7
center?
center.
)
-3
B) ‘3
B)
C) _§7
3
C)
A
0
A)) 660
B) 66
B) 66
D)
D) -‐22
C)) 7722
C
_
D
)
7
D) 78
8
the
what is the
of n, what
the same
is the
of 68 is
75% of
IfIf 75°/o
same as 85% of
vvalue
a l u eofdn?
"?
_
has 44
Jason has
cards, and
20 footba ll cards,
has 20football
Ian has
Ian
and Jason
44
that
such that
trade such
agree to trade
They agree
cards . They
baseball cards.
baseball
card
every card
cards for every
2 baseball cards
Ian 2baseball
gives Ian
Jason gives
Jason
trades
suc h trades
many such
how many
After how
Jason. After
Ian
gives to Jason.
Ian gives
number of
equa l number
an equal
have an
each have
and Jason
Ian and
will
Jason each
will Ian
cards?
cards?
|
|
_
A
)
9
A) 9
B
0
B)) 110
C)
11
C) 11
games .
their first 15
The
o n exactly
15 games.
exactly 4 of their
won
Pirates w
The Pirates
won
and won
games and
remaining games
They
played N remaining
then played
They then
the
all the
half of all
exactly half
won
all
o n exactly
they w
them. If they
all of them.
value of N ??
the value
what is the
played, what
games
they played,
games they
D) 12
D) 12
the
number x, the
the number
times the
from 3 times
IfIf 3 is subtracted
subtracted from
added to
when 8 is added
result when
the result
What is the
result
result is 21. What
half of x ?
A) 1
|
|
_
B
B) 5
)
5
C) 8
12
D) 12
D)
the same
with the
Alice
start with
same number
number of
Julie start
and Julie
Alice and
pens
her
of
16
gives
pens.
After
Alice
gives
16of
her
pens
to Julie,
Julie,
Alice
After
pens.
Alice
as
pens
Julie
w o times
many pens asAlice
as many
times as
has ttwo
then has
Julie then
the
have at the
Alice have
did Alice
pens did
does.
o w many
many pens
How
does. H
start?
start?
106
PANDA
COLLEGE PANDA
THE COLLEGE
i i
books on
on aa
the books
and g of the
Mark
w n %and
own
Kevin o
and Kevin
Mark and
than
less than
dollars less
a tie is k dollars
price of a
the price
store, the
a store,
At a
costs
shirt
a
If
shirt.
a
of
price
the price a shirt. If a shirt costs $40
times the
three times
three
value of k?
the value
what is the
costs $30, what
tie costs
and a tie
and
k?
the
rest of the
owns the
Lori owns
respectively . Lori
shelf, respectively.
shelf,
the rest
than
books than Mark,
more books
Kevin oowns
If Kevin
books . If
books.
w n s 9 more
Mark,
own?
Lori own?
does Lori
books does
many books
how many
how
has
rectangle has
a rectangle
shape of a
the shape
board in the
wooden board
A wooden
the
area of the
the area
If the
width. If
its width.
twice its
that is twice
a
length that
a length
length, in
feet, what
square feet,
board is 128 square
board
what is the
the length,
board?
the board?
feet, of the
feet,
grand
its grand
celebrate its
coupons to celebrate
out coupons
gave out
A bakery
bakery gave
either $1, $3,
opening.
Each coupon
coupon was
was worth
worth either
opening. Each
out
given out
were given
$1 coupons were
many $1coupons
as many
Twice as
or $5. Twice
coupons
$3
many
as
times
3
and 3 times as many $3 coupons
coupons, and
as$3
as $3 coupons,
value of
The total
coupons. The
as $5 coupons.
given oout
were
u t as$5
total value
were given
out was
coupons given
all
given out
was $360. How
H o w many
many
the coupons
all the
out?
given out?
were given
coupons were
$3 coupons
$3
A)) 440
A
0
B) 45
45
C)
48
C) 48
D
4
D)) 554
has
seas hells . Bob has
collect seashells.
all collect
Carl all
and Carl
Alex,
Alex, Bob, and
three
Alex ha
as Carl.
many seashells
half
seashells as
Carl. Alex
hass three
as many
half as
If Alex and
as Bob. If
times
and Bob
seashells as
many seashells
as many
times as
seashells
many seashells
how many
seashells, how
have 60 seashells,
together
together have
have?
Carl have?
does
does Carl
A water tank is connected to two pipes, Pipe A
and Pipe B. It takes 4 hours to fill the tank when
only Pipe A is in use, and it takes 6 hours to fill
the tank when only Pipe Bis in use . If it takes 111
minutes to fill the tank when both Pipe A and
Pipe Bare in use, what is the value of m ?
A
5
A)) 115
B) 20
20
C
0
C)) 330
D
D)) 4400
107
CHAPTER 12 WORD
CHAPTER
WORD PROBLEMS
PROBLEMS
Yoona runs
steady rate
yard per
second..
runs at a steady
rate of 1 yard
per second
Jessica runs
Jessica gives
runs 4 times
times as
as fast. IfIf Jessica
gives Yoona
head start
start of 30 yards
a head
yards in a race,
race, how
how many
many
yards must
must Jessica
Jessica rrun
yards
u n to catch
catch up to Yoona?
Terry is hired
hired to pave
pave a
a parking
parking lot
lot and
and finishes
finishes
1
3 of the
hired to
the parking
parking lot
lot before
before Andy
Andy is
is hired
to work
work
3
alongside him.
alongside
him . They
They each
each work
work at
at a
a constant
constant
rate , but
but Terry works
rate,
works twice as
as fast as
as Andy
Andy does.
does.
x
1 + i)1 )
2 can be used to
The equation
equation 9
9 (i
x = 2 can
( +
3 be used to
2
the total
total number
number of days
find the
days x it would
would have
have
taken Terry to pave
pave the
taken
lot by
by
the entire
entire parking
parking lot
himself. Which
Which of the
the following
himself.
the best
following is the
best
interpretation of the
the number
number 9 in the
interpretation
the equation?
equation?
A) The
The number
number of days
days it would
would have
have taken
taken
Terry and
and Andy
Andy to pave
pave the
parking
the entire
entire parking
they had
had worked
worked together
together from
from the
lot ifif they
the
start.
start.
number of days
days it will
will take
take Terry and
B) The number
and
Andy to pave
pave the
the remainder
remainder of the
the parking
Andy
parking
working together.
together .
lot working
The number
number of days
days it would
would take
take Andy
Andy to
C) The
pave the
the remainder
remainder of the
the parking
parking lot
pave
lot if
if he
he
were working
working alone.
alone .
were
The number
number of days
days it would
would take
take Terry
D) The
Terry to
pave the
the remainder
remainder of the
the parking
parking lot
he
pave
lot ifif he
were working
working alone.
were
alone.
Nicky owns a house that has a patio in the shape
of a square . She decides to renovate the patio by
increasing its length by 4 feet and decreasing its
width by 5 feet. If the area of the renovated patio
is 90 square feet, what was the original area of
the patio, in square feet?
108
13
Minim
Minimum
Maximum
um & Maxim
um
Word
Word Proble
Problems
ms
Minimum
Minimum and
and maximum
word problems
problems require
require aa bit
bit of
maximum word
of logic and
and an
an understanding
understanding of
of rates
rates and
and inequalities
inequalities
(chap
(chapters
and 11). One
One of the most
most common
common issues
students
have
is
that
they’re
unsure
of
whether
round
ters 4 and
issues studen
ts have is that they 're unsure of whether to
to round
up or down.
down. The examples
examples in this
chapter will address
this
issue
and
illustrate
the
strategies
you’ll
need
to
this chapter
address
issue and illustrate
strategies you' ll need to
solve
solve these types
types of problems.
problems.
EXAMPLE 1: Corinne
graphic designer
designer who
who earns
logo she
designs.. What
What is
is the
the
EXAMPLE
Corinne is aa graphic
earns $275 for
for every
every logo
she designs
minimum number
number of logos
logos she would
would have
have to design
design to earn
least $4,000
minimum
earn at
at least
$4,000 ?
To earn
Corinne would
would have
have to design
design at least
least 4,
To
T
earn at least
least $4,000,
$4,000, Corinne
\~
0
_
x 14.5
14.3 logos.
logos. That's
That’s 14
14 logos
logos and
and half
half aa
::::::
logo. But becau
because
it’s implied
implied that
that a fraction of a logo
logo cannot
up to
to
se it's
cannot be
be designed
designed and
and sold,
sold, we have
have to round
round up
ii logos.
OI)
logos.
When
When a whole
whole number
is implied,
implied, the
the minimum
m i n i m u m generally
that we
we round
up.
number answer
answer is
generally requires
requires that
round up.
EXAMPLE
EXAMPLE 2: A pallet
pallet truck
truck can
can move
move up
up to 3 tons
single trip.
be used
used to
to move
move
tons in a single
trip. If
If the
the truck
truck is
is to
to be
320-pound
320-pound pallets,
pallets, what
what is the
the maximum
maximum number
number of whole
pallets the
truck can
can move
move in
in aa single
single trip?
trip?
whole pallets
the truck
(1 ton
pounds)
ton = 2,000 pounds)
A) 6
A)6
B) 18
C) 19
D) 106
6,
000
6,000
: 18.75 pallets.
pallets. However,
However,
=
320
the question
question specifically
specifically states
states whole
whole pallets,
pallets, so
sowe
have to
E pallets
pallets.. If
rounded up,
up, the
we have
to round
round down
down to
to DI]
If we
we rounded
the
weight
would
weight would be
be above
above what
what the truck
truck can handle.
handle.
Since 3 tons is equivalent
equivalent to 3 x 2,
000 =: 6,000 pounds,
pounds, the truck can move
move
2,000
When
number answer
generally requires
When a whole
w h o l e number
a n s w e r is implied,
implied, the maximum
maximum generally
requires that we
we round
round down.
down.
109
MAXIMUM WORD
MINIMUM & MAXIMUM
CHAPTER 13 MINIMUM
CHAPTER
WORD PROBLEMS
PROBLEMS
what
yogurt, what
greek yogurt,
of greek
6 cups
and 6
flour and
of flour
cups of
8 cups
flatbread can
tray of flatbread
one tray
EXAMPLE 3: H
If one
can be
be made
made from
from 8
cups of
and 100
of flour
cups of
from 150
made from
be made
can be
that can
flatbread that
whole trays
number of whole
maximum number
the maximum
is
is‘the
trays of flatbread
150 cups
flour and
100 cups
cups
yogurt?
gteEtkyogurt?
of greek
other,
used up
be used
will be
yogu rt) will
greek yogurt)
resources (either
the resources
questions , one
In these
one of the
(either flour
flour or greek
up before
before the
the other,
types of questions,
these types
In
these
approach
to
way
best
the
Therefore,
produced.
be
can
that
amount
the
limit
resource will
and that
will limit the amount that can be produced. Therefore, the best way to approach these
that resource
and
resource separately.
each resource
consider each
questions is to consider
questions
separately.
10
the
consider the
only consider
we only
If we
made . If
be made.
can be
flatbread can
tra ys of flatbread
l~O = 18.75 trays
requirement, %
the flour
consider the
only consider
we only
IfIf we
flour requirement,
100
be produced
can be
that can
amount that
the amount
Since the
made . Since
be made.
can be
flatbread can
trays of flatbread
l~O ~
requirement, ‐6‐
yogurt requirement,
greek yogurt
greek
2 16.7 trays
produced
limiting
a limiting
is a
yogurt is
greek yogurt
the greek
flour, the
the flour,
produced from
can be
that can
amount that
the amount
than the
less than
from
greek yogurt
yogurt is less
beproduced
from the
the greek
from the
the
result,
a
As
.
flatbread
of
trays of flatbread. As a result, the
limited to 16.7 trays
flour, so
use up
enough of it to use
isn't enough
factor.
up the
the flour,
so we're
we’re limited
There isn’t
factor . There
when finding
round down
we round
that we
Remember that
made is [!I].
be made
can be
that can
trays that
number of trays
whole number
maximum
-.
Remember
down when
finding the
the
maximum whole
maximum.
maximum.
EXAMPLE 4:
EXAMPLE4:
= 18tw
18tw +
+ 1,050
1,050
C=
producing
of producing
dollars, of
in dollars,
C, in
total cost
the total
calculate the
equation above
uses the
appliance manufacturer
An appliance
manufacturer uses
the equation
above to calculate
cost C,
more than
no more
spend no
can spend
manufacturer can
the manufacturer
that ea-ch
toasters that
shipment of t toasters
a shipment
a
each weigh
weigh w
to pounds.
pounds. If
If the
than $21,000
$21,(X)0
the
what is
pounds, what
6 pounds,
be 6
will
toaster w
each toaster
and the
toasters, and
shipment of toasters,
producing
producing the
the next
next shipment
the weight
weight of each
i l l be
is the
shipment?
next
the
for
produced
be
can
that
toasters
of
number
maximum
maximum number toasters that can be produced
the next shipment?
up an
setting up
question by
this question
solve this
Let's solve
Let’s
by setting
an inequality:
inequality:
c ~
5 21,000
21,000
C
21,000
::S:21,000
18110 +
+ 1,050
1,050 5
lBtw
18t(6) ::S:
5 19,950
19,950
18t(6)
10815
19,950
::S:19,950
108t
184.72
::S:184.72
t5
produced
be produced
number that
maximum number
the maximum
numbers , the
whole numbers,
produced in whole
are produced
toasters are
that toasters
implied that
it's implied
Since it’s
Since
that can
can be
I184 1.
for the
shipment is -.
next shipment
the next
110
THE
COLLEGE PANDA
THE COLLEGE
PANDA
EXAMPLE
deck of48cards
of 48 cards consists
EXAMPLE 5:
5: A
Adeck
consists of
ofonlynedcards
Ifthenumberofredcards
only red cards andblackcards.
and black cards. If
the number of red cards
is less
than twice
twice the
the number
less than
number of black
what is the
in the
black cards,
cards, what
the minimum
minimum possible
possible number
number of
of black
black cards
cards in
the
deck?
deck?
Let r be
be the
the number
Let
number of red
red cards
cards and
and b be
the number
variables, we
can set
be the
number of black
black cards.
cards. Using
Using these
these variables,
we can
set up
up a
a
system
that consists
consists of an
system that
an equation
and an
an inequality.
equation and
inequality.
r+b=
: 448
8
b
r <<22b
Since the
the question
Since
question is asking
asking about
about the
black cards,
cards, our
o u r goal
get rrid
i d of
the black
goal should
should be
be to
to get
of rr so
so that
that we
we end
end up
up with
with an
an
inequality
inequality in terms
terms of bonly.
so, we
we isolate
isolate rr in
in the
o w we
48‐- bb
b only. To do
do so,
the equation
equation to
to get
get rr = 48‐
48 - b.
b. N
Now
we substitute
substitute 48
for r in the
inequality :
the inequality:
48
4 8-‐ bb <<22b
b
4
b
488 < 33b
l166 < b
Based on
on this
this resulting
resulting inequality,
inequality, the
Based
minimum possible
possible number
black cards
the minimum
number of
of black
cards is-.
is [ill.
EXAMPLE
EXAMPLE 6: An art
art teacher
teacher needs
needs to buy
buy aa total
of 36paintbrushes
Each paintbrush
total of
36 paintbrushes for a
a painting
painting class.
class. Each
paintbrush
must
an acrylic
acrylic brush,
brush, which
must be
be either
either an
which costs
brush,
which
costs
$3.
If no
no more
more than
than
costs $5, or a watercolor
watercolor brush, which costs
$150 can
can be
spent
on the
be spent on
the paintbrushes,
paintbrushes, what
what is the
minimum number
watercolor brushes
brushes the
the art
art teacher
the minimum
number of
of watercolor
teacher
can buy?
can
buy?
be the
the number
number of acrylic
Let a be
acrylic brushes
brushes.. Now
Now we
we can
can set
set up
up aa
brushes and
and w be
be the
the number
number of watercolor
watercolor brushes
system of an
equation and
system
an equation
inequality
just
aswe
did
in
Example
5.
and an
an inequality just as we did in Example 5.
a+w
6
a+
w ==336
5 a+
+ 33w
w §:S1150
50
5a
Our goal
get the
Our
goal is to
to get
the inequality
inequality in terms
w only,
so let’s
a in
in the
equation to
to get
get a
z 36
36‐- w.
w. Now
Now
terms of w
only, so
let's first isolate
isolate a
the equation
a=
we
can substitute
we can
substitute for a
a in the
the inequality:
inequality :
5(36 -‐ w)
w) +
+ 3w
$ 1150
50
5(36
3w :S
180 -‐ 510
5w + 310
3w 5150
:S 150
180 ‐- 2w 5
:S 150
‐2w
5 ‐30
- 2w :S
- 30
w
215
w ~
15
Based on
on this
this resulting
resulting inequality,
Based
m i n i m u m number
can be
bought is-.
inequality, the
the minimum
number of watercolor
watercolor brushes
brushes that
that can
be bought
is lli].
111
111
CHAPTER 13 MINIMUM
CI-LAPTER
MINIMUM &
& MAXIMUM
MAXIMUM WORD
WORD PROBLEMS
PROBLEMS
EXAMPLE 7: Shahar
baseball cards
Two rare
rare
Shahar collects baseball
cards that
that are
are sold
sold in
in regular
regular packs
packs and
and premium
premium packs.
packs. Two
cards can be found inevery
cardscanbefound
regularpackand
threerarecardscanbeformd
ineverypremium
pack.
in every regular
pack and three rare cards can be found in every premium pack. If
If
Shahar wants
Shahar
wants to add
30
rare
cards
his collection
add at least
least 30 rare cards to
to his
collection by
by buying
buying no
no more
more than
than 12
12 packs
packs of
of baseball
baseball
cards, what
cards,
what is the
premium packs
packs he
the least
least number
number of
of premium
he could
could buy?
buy?
Again, let’s
let's set up
Again,
up a
a system
system with
with r as
as the
the number
number of
of regular
regular packs
packs and
and pas
pas the
the number
number of
of premium
premium packs.
packs .
r+
+ pp §:S1122
2
0
2rr + 3
3pp 2
~ 330
Since we
we have
have a
a system
system of ttwo
inequalities, we can’t
w o inequalities,we
can't just
just do
do what
what we
we did
did in
in Examples
Examples 5
5 and
and 6.
6. Instead,
Instead, we
we need
need
on the following
following trick: inequalities
inequalities can be
to rely on
same
direction.
be added
added together
together if
if their
their signs
signs point
point in
in the
the same direction.
Note that
that inequalities
inequalities should
should never
never be
be subtracted
Note
subtracted from
from one
one another;
another; only
only think
think in
in terms
terms of
of addition.
addition. Soto
So to get
get
signs to point in the
the same
same direction,
direction, we can multiply the first inequality
the signs
‐
2
.
This
will
switch
inequality by
by - 2. This will switch the
the sign
sign
and get
get the
the coefficients
coefficients of r to match
and
match up.
up .
-‐ 22rr -‐ 22p
p2
~ ~24
- 24
2r+3p
2r + 3p 2~ 30
30
Now we
we can
can add
add the inequalities
Now
inequalities together
together to
to get
p ~> 6
p
6
Based on this result,
result, the minimum
minimum possible
Based
value of
p is [E].
possible value
of pis
[ill.
Another valid
valid way
way to approach
approach this problem
problem is guess
Another
guess and
and check.
check . For example,
example, we
we can start
start with
with p
p =
=0
0 and
and
Given those
those values,
values, is 2r +
least 30? If not,
rr =
z 12. Given
+ 3p at least
with
p
z
1
and
r
=
11,
and
not, repeat
repeat the process
process with p = 1 and r = 11, and etc.
etc.
Soon enough,
enough, you’ll
you'll arrive
arrive at p =
Guess and
and check turns
Soon
z 6. Guess
turns oout
u t to be
be quite
quite efficient in
in many
many cases,
cases, so
so don’t
don't
give up
up on
on it too early.
112
THE COLLEGE
COLLEGE PANDA
THE
PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 304.
A calculator is allowed
allowed on the following
following
questions.
questions.
shop held
held a
a weekend
weekend sale
the goal
A gift shop
sale with
with the
goal of
selling at
at least
least $8,000 worth
greeting cards
cards
selling
worth of greeting
and gift
boxes . Each
greeting card
was sold
sold for
and
gift boxes.
Each greeting
card was
more
and each
each gift
box was
was sold
$5, and
gift box
sold for $7. If no more
than 400 gift
gift boxes
were sold
the sale
due
than
boxes were
sold during
during the
sale due
limited inventory,
inventory, what
the minimum
to limited
what is
is the
minimum
number of greeting
greeting cards
cards the
could have
have
number
the shop
shop could
sold to meet
meet its
goal?
sold
its goal?
Katherine
classroom calculators
each
Katherine has
has 28
28classroom
calculators that
that each
require aa set
require
set of 4
4 batteries.
batteries. If
If her
her school
school supplies
supplies
her with
batteries in packs
what is the
her
with batteries
packs of 6, what
the least
least
number of packs
to provide
provide every
number
packs needed
needed to
every
classroom
with aa complete
classroom calculator
calculator with
complete set
set of
batteries?
batteries?
A) 1,040
B) 1,160
C) 1,280
D) 1,400
restock supplies,
supplies, a
a nail
salon purchases
To restock
nail salon
purchases
toolkits that
that each
80 nail
and 150
toolkits
each include
include 80
nail files and
the nail
nail salon
restock at
nail buffers.
buffers. If the
nail
salon needs
needs to restock
at
least 1,800 nail
at least
nail
least
nail files
files and
and at
least 4,000 nail
buffers, what
what is the
toolkits
buffers,
the minimum
minimum number
number of toolkits
the salon
salon can
can purchase?
purchase?
the
Martha
Martha is
is working
working on
on aa design
design project
project that
that
requires
requires 16
16ounces
ounces of glue.
glue. The
The glue
glue gun
gun she
she is
is
using
comes
preloaded
with
a
glue
stick
that
using comes preloaded with a glue stick that
provides 2.5 ounces
ounces of glue.
provides
glue. The
The only
only additional
additional
glue sticks
purchase are
glue
sticks Martha
Martha can
c a n purchase
are ones
ones that
that
each
each provide
provide 1.75 ounces
ounces of glue
glue.. Assuming
Assuming that
that
the
the glue
glue sticks
sticks can
can only
only be
be purchased
purchased in whole
whole
numbers,
minimum number
numbers, what
what is the
the minimum
number of glue
glue
sticks Martha
Martha must
must purchase
purchase for
for her
her project?
project?
sticks
A ) 6
A)
B)
B)7
C)8
C)
D
D)) 9
One
equivalent to approximately
approximately 33.8
One liter
liter is equivalent
ounces.
can each
ounces. Mark
Mark has
has plastic
plastic cups
cups that
that can
each hold
hold
12
liquid.. At most,
12 ounces
ounces of liquid
most, how
how many
many of these
these
plastic Cups
cups could
could aa ttwo
liter bottle
bottle of soda
plastic
w o liter
soda fill?
A)5
A)
B
B)) 6
C) ) 7
C
D
D) ) 8
113
CHAPTER 13 MINIMUM & MAXIMUM WORD PROBLEMS
During a
a weekolong
week-long fishing
fishing trip,
trip, Ashleigh
During
Ashleigh
caught nine
less than
than three
three times
the number
nine less
times the
number of
caught
Naomi caught.
caught. If they
45 fish
fish Naomi
they caught
caught at
at least
least 45
combined, what
the minimum
minimum number
number of fish
fish
combined,
what is the
that Naomi
could have
have caught?
caught?
that
Naomi could
In one hour, Jason can install at least 6 windows
but no more than 8 windows. Which of the
following
a possible
following could
could be
bea
possible amount
amount of time,
time, in
hours, that
windows in
hours,
that Jason
Jason takes
takes to
to install
install 100 windows
home?
a home?
A)
12
A) 12
B)
16
B) 16
C)
17
C) 17
D) 18
18
IN
1 fluid
fluid ounce
ounce =
= 29.6 milliliters
milliliters
I’
'
'
F
V
filled with
with black
white pebbles,
pebbles,
A jar is filled
black pebbles,
pebbles, white
and jade
pebbles. The
pebbles is
and
jade pebbles.
The number
number of jade
jade pebbles
greater than
than half
half the
the number
number of black
greater
black pebbles,
pebbles,
and the
the number
number of white
white pebbles
pebbles is less
less than
than
and
twice
the number
number of black
black pebbles.
pebbles . If
there are
are
twice the
If there
32 jade
the jar, what
what is the
32
jade pebbles
pebbles in the
the maximum
maximum
number of white
white pebbles
pebbles that
that could
could be
jar?
number
be in the
the jar?
1 cup
cup =
= 16 fluid
fluid ounces
ounces
A chemistry
teacher is planning
chemistry teacher
planning to
to nm
r u n aa class
class
experiment
each student
experiment in which
which each
student must
must measure
measure
out
a graduated
out 100
100 milliliters
milliliters of vinegar
vinegar in a
graduated
cylinder.
class is
using 66 cups
cylinder. The
The class
is ljmited
limited to
to using
cups of
vinegar. Given
Given the
the information
above, what
vinegar.
information above,
what is
the maximum
who will
the
maximum number
number of students
students who
will be
be
able
participate in this
experiment?
able to participate
this experiment?
Giovanni
Giovanni works
works as
as aa waiter
waiter at
at an
an Italian
Italian
restaurant.
For
every
table
that
he
restaurant. For every table that he serves,
serves, he
he
earns
the bill.
earns a 15%
15% tip
tip on
on the
bill. During
During lunch,
lunch, he
he
served
tables and
table had
served 12 tables
and each
each table
had an
an average
average
bill of
during dinner
bill
of $25. If each
each table
table during
dinner will
will have
have
an
bill of $45, what
least number
an average
average bill
what is the
the least
number
Giovanni must
must serve
during dinner
of tables
tables Giovanni
serve during
dinner to
earn
least $180 for the
the day?
earn at
at least
day?
A) 3
A)
B
6
B)) 116
C
8
C)) 118
D ) 2200
D)
114
s
THE
COLLEGE PANDA
PANDA
THE COLLEGE
part of aa marketing
marketing campaign,
As part
campaign, aa restaurant
restaurant is
offering 4
4 free
free tacos
tacos for every
every burrito
burrito a
a customer
customer
offering
buys. If
If the
the restaurant
restaurant would
the
buys.
would normally
normally sell
sell the
tacos for $2.60 each,
what is
is the
the minimum
minimum
tacos
each, what
number of burritos
burritos aa customer
customer would
would have
have to
number
to
buy to receive
receive at
at least
least $140 worth
buy
worth of tacos
tacos for
free?
A pharmacy
produces a
a certain
pharmacy produces
certain medication
medication in aa
daytime
daytime variety
variety and
and aa nighttime
nighttime variety
variety.. A bottle
bottle
of the
the daytime
daytime variety
variety contains
contains 2 ow1ces
ounces of the
the
active
ounces of flavored
active ingredient
ingredient and
and 66 ounces
flavored syrup
syrup..
A bottle
bottle of the
the nighttime
nighttime variety
variety contains
contains 33
ounces
active ingredient
ingredient and
ounces of the
the active
and 55 ounces
ounces of
flavored syrup.
flavored
syrup. The
The pharmacy
pharmacy currently
currently has
has no
no
more
than 385 ounces
the active
more than
ounces of the
active ingredient
ingredient
and
and no
no more
more than
than 850 ounces
ounces of flavored
flavored syrup
syrup
available
at least
65 bottles
the daytime
available.. If at
least 65
bottles of the
daytime
variety
filled, what
variety must
must be
be filled,
what is the
the maximum
maximum
number
of
bottles
of
the
nighttime
number bottles the nighttime variety
variety that
that
can
filled?
can be
be filled?
A)
A ) 778
8
B) 85
C)
92
C) 92
decorating two-tier
two-tier and
and three-tier
Ava is decorating
three-tier
wedding
cakes.
It
takes
her
20
minutes
to
wedding cakes. takes her 20 minutes to
decorate each
each two-tier
two-tier wedding
cake and
and 35
decorate
wedding cake
35
minutes to decorate
decorate each
each three-tier
minutes
three‐tier wedding
wedding
cake . If
decorate at least
cake,
If Ava needs
needs to decorate
least 14
wedding cakes
cakes today,
today, and
she can
no
wedding
and she
can spend
spend no
more than
than 6 hours
doing so,
so, what
the
more
hours doing
what is the
maximum
three-tier wedding
wedding cakes
cakes
maximum number
number of three-tier
she can
decorate today?
today?
she
can decorate
D) 106
A banquet
banquet hall
hall has
has aa maximum
maximum seating
seating capacity
capacity
of 168 people
particular event,
people.. For
For aaparticular
event, the
the banquet
banquet
manager
manager must
m u s t use
use an
an arrangement
arrangement of short
short
tables and
that there
there is
and long
long tables
tables to ensure
ensure that
tables
enough
that capacity.
capacity. Each
enough seating
seating to meet
meet that
Each short
short
table seats
table
seats 4
4 people
people and
and each
each long
long table
table seats
seats 8
8
people
.
lf
no
more
than
32
tables
can
be
placed
people. If no more than 32 tables can be placed
inside
the banquet
banquet hall,
maximum
inside the
hall, what
what is the
the maximum
number
tables that
be used?
number of short
short tables
that can
can be
used?
A
A)) 4
B)5
B)
C
C) ) 6
D
D) ) 8
A ) 110
0
A)
B)
B) 14
14
C) 18
C)
18
D
D)) 2222
115
CHAPTER 13 MINIMUM & MAXIMUM WORD PROBLEMS
Lianne wants to make a seasoning that consists
of 75% sea salt and 25% black pepper . If sea salt
costs $2 per pound and black pepper costs $8 per
pound, and Lianne can spend no more than $210
on these ingredients, what is the maximum
number of pounds of seasoning that she wiU be
able to make?
C
10On 0/
= 100n
%
_n+w
n+w
0
The formula
determine the
formula above
above can
can be
be used
used to determine
the
volume percent
volume
concentration
C
of
an
ethanol
percent concentration
an ethanol
solution
ethanol and
w
solution containing
containing 11
n ounces
ounces of ethanol
and w
ounces
wants to use
ounces of water.
water . A chemist
chemist wants
use the
the
formula
formula to create
create an
an ethanol
ethanol solution
solution with
with aa
volume percent
volume
than
percent concentration
concentration of no more
more than
16%. If
will mix 10
10 ounces
ethanol
If the chemist
chemist will
ounces of ethanol
and x cups
and
water to create
cups of water
create the
the desired
desired
solution,
solution, what
the minimum
minimum possible
value of
what is the
possible value
assuming that
x, assuming
that x is aa whole
whole number?
number?
(1 cup
= 8 ounces)
cup =
ounces)
A)) 442
A
2
B) 50
50
B)
C)) 556
C
6
D)) 6600
D
its products
small,
A toy company
company ships
ships its
products in small,
medium
,
and
large
boxes
.
Last
month, the
the
medium, and large boxes. Last month,
company shipped
a total
total of 250 boxes,
company
shipped a
boxes, of which
which
70 were
were medium
medium boxes.
boxes. The
The number
number of large
70
large
boxes shipped
shipped was
was more
more than
than the
the sum
sum of the
boxes
the
number of small
small boxes
boxes shipped
shjpped and
and the
the number
number
number
medium boxes
boxes srupped.
What is the
of medium
shipped. What
the greatest
greatest
possible number
number of small
boxes the
company
possible
small boxes
the company
shipped last
shipped
last month?
month?
t
.
i
116
Lines
functions .
as linear
referred to as
often referred
they are
Lines are
just functions
functions in the
f(x) =
: mx +
+ b, which
which is why
why they
are often
linear functions.
form of J(x)
the form
are just
Lines
present
they
because
first
Lines
covering
we're covering lines
We'll cover
whole in aa future
future chapter;
chapter; we’re
because they present some
some
a whole
as a
functions as
cover functions
that they
frequently that
so frequently
concepts so
these concepts
tests these
The SAT tests
functions . The
concepts
don’t apply
they deserve
deserve their
their
other functions.
apply to other
that don't
concepts that
o
w n chapter.
chapter. Let's
Let’s dive
dive in!
own
Given
w o points
and (x2,Y2
(Md/2)) on aa line,
line,
(x1,yi) and
points (x1,_1/1)
any ttwo
Given any
,
rise
2 -~ y1
1/2
rise
· -- “ll‐1
Slope
o f line
ope of
SI
Line =: -4 =z -y
rrun
un
.\'
3
XJ
X2 -~ x1
The rise
line is. The
more
the m
slope, the
the slope,
The
measure of the
the steepness
steepness of aa Line-the
line‐the bigger
bigger the
o r e steep
steep the line
rise is the
the
a measure
slope is a
The slope
of
slope
A
coordinates.
x
the
between
distance
the rrun
and the
coordinates and
they.'1 coordinates
distance
between the
u n is the
the distance between the coordinates. slope 2 means
means
distance between
to the lleft.
every 1 uunit
down
the line goes
goes 2 uunits
n i t s up
up for every
u n i t to
to the
the right,
right, or 2 uunits
nits d
o w n for every
n i t to
e f t , A slope
slope of -‐ ~5
every 11 unit
the left.
to the
units to
every 3 units
up for every
units up
or 2 units
right, or
down for every
means the
the Line
line goes
goes 2 units
units down
every 3 units
units to
to the right,
left.
means
the graph
as in the
go up
always go
slope always
Lines
up and
and to the
the right
right as
graph above.
above.
positive slope
with positive
Lines with
yV
rise
run
the right
and to the
down and
Lines
right::
go down
slope go
negative slope
with negative
Lines with
y
117
CHAPTER
CHAPTER 14
14 LINES
LINES
EXAMPL£1~
EXAMPLE1:
y
(a.b)
The line
line shown
the xy-plane
passes through
through the
the origin
and point
point (a,
b), where
where a > b. Which
shown in the
xy-plane above
above passes
origin and
(a,b),
Which of
the following could
bethe
the slope
slope of the
the line?
line?
thelollowing
could be
1
A)
-A)‐-%
2
B);~4
B)
C ) 11
C)
D);
D)~
2
rise ·_ 1
b
h at tthe
h esslope
lope 1s
.. positive.
.. .. Th
Iope, -rise
.. tthat
..
F
First,
notice
lS
posnhve.
The
ru_', 1s
15aalso
1rst,
notice
esslope,
so equal
equa I to E'
-b .
runI]
a
y
Since a > b,
b, ~g is always
less than
than 1.
1. For
For example,
example, if
if aa =z 5 and
be ~2 The
The only
choice
Since
always less
and b = 3,
3, the
the slope
slope would
would be
only choice
5
a
[NJ.
that’s both
both po
positive
less than
than 11 is
is answer
answer (B) .
that's
sitive and
and less
EXAMPLE 2:
2: Line
Line m
m passes
passes through
points (k, 7) and
and (3,1:
4). If
If the
the slope
line m
m is 3, what
what is the
EXAMPLE
through points
(3, k -‐ 4).
slope of line
the
value of k?
k?
value
( k- 4)
S l o =p ---e(k
z3 ‐-43k)-‐T7 == 33
Slope
1 == 33(3
( 3 -‐ kk))
k -‐ 111
2 99 ‐- 33kk
k - 1111 =
4 k=
: 2200
4k
k:‑
118
118
THE
COLLEGE PANDA
THE COLLEGE
PANDA
EXAMPLE
EXAMPLE 3:
3:If
line has
has a
and passes
passes through
through the
the point
) , which
If a
a line
a slope
slope of
of~g and
point (1,
(1, ‐- 22),
which of
of the
the following
following points
points
also
also lies
lies on
on the
line?
the line?
A) ((-2,
A)
4 , --5)
5)
B)(-‐2,‐1)
B) (-2,-1)
cue‐11)
C) (4, -1)
D) (4,
(4,10)
D)
10)
i
1
A slope
to the
right, or
or 11 down
down for
slope of 5 means
means 11up
up for every
every 33 to
the right,
for every
every 3
3 to
to the
the left.
left. If
If we
we go3
go 3 to
to the
the left,
left, the
the point
point
[I§J.
we get
we
If we
we go
go3
to the
the right,
right, the
the point
we get
get to
the line
line is
is ((4,
answer (C) .
get to
to on
on the
the line
line is
is ( -‐ 22,, -‐3).
3). If
3 to
point we
to on
on the
4, -~1),
1), answer
In this
we got
the answer
answer pretty
quickly, but
but if we
we hadn’t,
would have
continued moving
moving right
or
this case,
case, we
got to the
pretty quickly,
hadn't, we
we would
have continued
right or
left
left until
until we
we fotmd
found an
answer choice
choice that
that matched
matched.. On
On the
an answer
the SAT, it shouldn't
shouldn't ever
ever take
take too
too long
long to
to arrive
arrive at
at the
the
answer for aa question
answer
question like
like this
this..
In addition
slope , you
addition to slope,
you also
know what
what x and
and y intercepts
The x-intercept
x-intercept is
the graph
graph
also need
need to know
intercepts are.
are. The
is where
where the
crosses the
the x-axis. Likewise,
crosses
Likewise, they-intercept
the y-intercept is where
where the
the graph
y‐axis..
graph crosses
crosses the
the y-axis
Let's say
say we
we have
Let’s
have the
the line
line
2x + By
2x
3y = 12
12
=
the x-intercept,
To find the
x‐intercept, set
equal to 0.
set y equal
2x +
+ 3(0)
3(O) =
= 12
2x
: 12
2x =
12
x =
: 6
The
The x-intercept
x-intercept is 6.
To find the
the y-intercept,
y-intercept, set
equal to 0.
0.
set xx equal
2(0) + 3y =
12
= 12
3];
3y := 12
12
y =z 4
The
y‐intercept is 4.
They-intercept
+
EXAMPLE
the line
line ax
ax + 3y = 15,
where aa is a
EXAMPLE 4:
4: If the
15, where
x‐intercept that
is twice
twice the
value of
a constant,
constant, has
has an
an x-intercept
that is
the value
of
the
y-intercept,
what
the y-intercept, what is
is the
the value
value of aa ??
First,
set x =
First, set
= 0Oto
find the
y-intercept:
to find
they-intercept:
a ( 0 )++3y
3 y=: 115
5
a(O)
3
5
3yy = 115
yy =
=5
They-intercept
The y-intercept is 5,
5, which
x-intercept must
0,
which means
means the
the x-intercept
must be
be 5
5x
x2
2=
= 10.
10. Plugging
Plugging in
in x
x z
= 10,
10, y
y=
= 0,
a(10) +
+3(0)
: 15
a(lO)
3(0) =
15
10a
= 15
a =~
119
CHAPTER 14
CHAPTER
14 LINES
LINES
All
be expressed
All lines
lines can
can be
expressed in slope-intercept
slope-intercept form:
y = mx
mx +
+ bb
where m
the slope
they-intercept.
the line
the slope
y-intercept is
where
m is
is the
slope and
and b is the
y‐intercept. So
So for
for the
line y =
= 2x
2x ‐- 3,
3, the
slope is 2 and
and the
the y‐intercept
-‐ 33::
y
While
expressed in slope-intercept
it'll take
take some
work to
there . If you’re
you're
While all
all lines
lines can
can be
be expressed
slope‐intercept form,
form, sometimes
sometimes it’ll
some work
to get
get there.
given
y easy
easy to get
get the
the equation
the line.
line . But
But what
given aa slope
slope and
and aa y-intercept,
y-intercept, then
then of course
course it's
it’s reall
really
equation of the
what if
we're
a y-intercept?
y-intercept? Then
it'll be
more convenient
convenient to use
use
we’re handed
handed aa slope
slope and
and aa point
point instead
instead of aa slope
slope and
and a
Then it’ll
be more
point-slope form:
point-slope
form:
m(x
yy ~- yY1
i =m
( x-‐ xxi)
i)
where
point. For
For example,
let's say
want to find the
the equation
equation of aa line
line that
has aa slope
slope
where (x1,y
(x1,y1)
the given
given point.
example, let’s
say we
we want
that has
1 ) is the
point (1, -‐ 22).
the line
line is then
then
of 3 and
and passes
passes through
through the
the point
) . The
The equation
equation of the
yy -‐ ( (-‐ 22)) =
- 11))
= 33(x
(x~
Once
it’s in point-slope
form, we
we can
can then
then expand
expand and
and shift
shift things
around to get
form if
if we
we
Once it's
point -slope form,
things around
get to slope-intercept
slope-intercept form
need to.
need
( x-‐ 1)
l)
y -‐ ((-‐ 22)) =
= 33(x
= 33xx -‐ 3
y +2 =
y=
=33xx ~- 55
EXAMPLE 5:
EXAMPLES:
y
........
............ 'J
,_
,
.........
,_
,_
,.,.
-14. - SI
.........
2 - 1P
"'
i-....
I"
-X
'
Which of the
following could
could be
be the
the equation
of the
the line
line shown
in the
the xy-plane
above?
Which
the following
equation of
shown in
xy-plane above?
A ) y == ‐- Z2x+3
x+3
A)y
1
B)y=%x+3
B)y
= -2 x + 3
1
C)y=‐%x+3
C)y
=--x+
3
2
D ) y=
= 2x
2 x-‐ 33
D)y
To get
get the
the equation
of the
the line
: mx
mx +
+ b, we
we need
need to
to find the
y-intercept b,
The line
To
equation of
line y =
the slope
slope m
m and
and the
they-intercept
b. The
line crosses
crosses
the
y‐axis
at
3,so
b
:
3.
The
line
goes
downward
from
left
to
right,
down
1
for
every
2
to
the
right,
so
the y-axis at 3, sob = 3. The line goes downward from left right, down
every to the right , so the
the slope
slope
1
m is -‐ ~.
5. Therefore,
Therefore, the
the equation
equation of the
the line
line is y =
z -‐ ;%x+
x + 3. Answer
Answer ~-(C) .
120
THE
THE COLLEGE
COLLEGE PANDA
PANDA
EXAMPLE
and (3, 13).
1;3),What is the y‐intercept
y~interceptof Imel?
EXAMPLE6:
6: A line
line l1passes through
through the points
points (( - 22,3)
,3) and
line I ?
yz‐y1 = 1
Slope
=
133-‐ ‐ 33 =: 2
5
= 1 Y2- Y1 =
oPe X2
x 2-‐ xX11 33 ‐- ( -‐ 22)) 2
Using
-slope form,
Using point
point-slope
form, our
o u r line
line is
y -‐ 13
13 = 2(x -‐ 3)
Note
that we
3). The result
result will
will ttum
Note that
we could've
could’ve used
used the other
other point
point (( ‐ 22,,3).
u r n out
out to be
be the
the same.
same.
y -‐ 13
13 =: 2(x
2(x -‐ 3)
3)
y=
=22xx ‐- 66++113
3
y=
: 22x
x+
7
+
After
can easily
easily see that
-intercept is [z].
After putting
putting the equation
equation into
into slope-intercept
slope‐intercept form,
form, we
we can
that they
the y-intercept
..
There
There are
are aa few more
more things
things you
you need
need to know
know about
about lines.
lines.
Two lines
lines are parallel
parallel if they
they have
have the same
same slope.
slope.
y
Two lines
In other
other words,
one slope
Two
lines are
are perpendicular
perpendicular if
if the
the product
product of
of their
their slopes
slopes is
is -‐ 11.. In
words, ifif one
slope is
is the
the negative
negative
1
).
reciprocal
reciprocal of the
the other
other (e.g.
(e.g. 22 and
and -‐ E).
2
y
121
CHAPTER
CHAPTER 14
14 LINES
LINES
EXAMPLE
a slope
slope of ~g and
through the
point (4,
(4, 3).
line n
is perpendicular
EXAMPLE 7: Line
Line m
m has
has a
and passes
passes through
the point
3). If line
1:is
perpendicular to line
line
m
and passes
following could
could be
equation of linen?
m and
passes through
through the
the same
same point
point (4,3),
(4,3), which
which of the
the following
be the
the equation
line n ?
A)y=‐§x+9
= -- 32 x + 9
A)y
3
= - -x
- 3
B)y=‐§x‐‐3
2
B)y
3
3
2
D)y
=- -x + 9
D)y=‐%x+9
C)y
=- -x + 6
C)y=‐?2‐x+6
2
3
Because
perpendicular to line
~- Using
Because it's
it's perpendicular
line m,
m, linen
line Hmust
must have
have aa slope
slope of -~‐5'
Using point-slope
point-slope form,
form,
3
(x - 4)
y‐3z‐g(x‐4)
2
y - 3 =-
3
yz‐gx+6+3
2x + 6 + 3
y =-
3
yy=‐%x+9
=- x+ 9
2
We
answer is ~-( D ) .
Weget
get the equation
equation into
into slope-intercept
slope-intercept form
form to see
see that
that the
the answer
Finally, you'U need
lines . The
the vertical
that
Finally,you’ll
need to know
know the equations
equations of horizontal
horizontal and
and vertical
vertical lines.
The equation
equation of the
vertical line
line that
passes
through (3,
0) is x = 3.
passes through
(3,0)
3.
y
x=3
X
+-+--+---+---+--+--+--+--f-1-+
equation of the
the horizontal
horizontal line
line that
passes through
through (0,3
(0, 3)) is y =
: 3.
3.
The equation
that passes
yll
y=3
+-+--+---+---+--+---+--+---t---,1--+
122
X
THE
THE COLLEGE
COLLEGE PANDA
PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 307.
A calculator
calculator should
should N
O T be
be used
on the
the
NOT
used on
following questions.
following
questions.
ln the
In
(‐3,5) and
and (6,8)
(6, 8) lie
lie on
on
the xy-plane,
xy-plane, points
points (-3,5)
line
points
is
also
on
line 1.
I. Which
Which of
of the following
following points is also on
lline
i n e ]I??
What
What is the equation
equation of
of the
line parallel
to the
the line
parallel to
the
y~axis
right of
y-axis?
y-axis and
and 3
3 units
units to the
the right
of the
the y-axis?
A) (0,6)
(0, 6)
A)
B) (3,8)
C) (9,10)
C)
(9, 10)
D)
(12,11 )
D) (12,11)
A)) Xx=
A
= -‐ 3
B)
B ) x=3
x=3
C)) yy =
C
= ‐- 3
D
D)) yy== 3
y
y
(5, n)
(-1, 1)
•
+----, 0+-----.
•
X
Note:
Note: Figure
Figure not
n o t drawn
to scale.
scale.
drawn to
-
In the
the figure
figure above,
above, the
the slope
slope of
of the
the line
line through
through
The graph of LineI is shown in the xy-p lan e
above. Which of the following is an equation of
a line that is parallel to line I ?
the
t w o plotted
value
the two
plotted points
points is
is~%- What
What is
is the
the value
o
off n
n??
A) 9
2
A)y=‐§x+2
A) y =-- x + 2
3
B) 4
C) 3
2
B)y=§x+10
B) y = x + 10
3
7
D) 3
3
3
C)yzgx‐4
C) y = - x - 4
2
D
=3
D)) yy =
3xx-~ 11
xy-plane,
line has
has an
anx-intercept
of -‐22
In the xy-p
lane, a
a line
x-intercept of
and a
and
slope of
of the
the
a y-intercept
y-intercept of -‐ 44.. What
What is
is the
the slope
line?
line?
A
A)) -‐ 2
B) ~§11
B)
-6
2
1
C)
C) -5
2
D) 2
123
CHAPTER 14
LINES
y
y
X
+---+-----+---<>----+-----
-3 -2 - 1 O
1
2
3
xy-coordinate system
system above
can be
be
Line I/ in the xy-coordinate
above can
represented by the
b. Which
represented
the equation
equation y = mx
mx + b.
Which
of the
the following
following must
must be
be true?
true?
the xy‐plane
xy-plane above,
In the
the graph
graph of the
the linear
linear
above, the
function
f is perpendicular
perpendicular to the
graph of the
the
function J
the graph
linear
(not shown).
shown). If
If the
graphs of
linear function
function g (not
the graphs
offf
and g intersect
the point
and
intersect at the
point
A) mb
A)
mb >
>0
B)
B) mb
mb < 00
(1,
2), what
what is the
the
(1,~),
C) mb
= 0
mb =
value of gg(‐1)
value
(- 1) ?
?
D) mb
: 1
mb =
The line y = -‐ 22xx -‐ 22 is perpendicular
perpendicular to line/.
line I. If
If
these two lines have the same y-intercept, which
of the following could be the equation of line / ?
A
A)) yy=z ‐- Z
2xx-‐ z2
B
2 x- ‐ 2
B) ) yy== 2x
A calculator is allowed
allowed on the following
following
questions.
questions.
1
C)
C) .v yz‐ix‐Z
=-- 2 x - 2
1
D)y=%x‐2
D) y = - x - 2
2
y
m
The slope
slope of line
line/ I is ~%and
and its
its y‐intercept
y-intercept is 3.
What
What is the equation
equation of the
the line
line perpendicular
perpendicular to
line
(1,5)
line I that
that goes
goes through
through (1,
5) ??
A
= ‐ 22xx + 3
A)) yy =B
= ‐ 2x
h+
B) ) yy =+7
What
m in the
figure
What is the slope
slope of the
the line
line m
the figure
above?
above?
A)
A) - 2
1
11
11
C)
C) y y:_§x+7
= - 2x + 2
1
B
B) ) ‐ ‑
1
9
D)
D) yy‐§x+§
= 2x + 2
2
1
C) C)
4
1
D) 2
124
THE
COLLEGE PANDA
PANDA
THE COLLEGE
2
A line
line with
with aa slope
passes through
slope of g passes
through the
the
In the xy-plane, the line with equation
1
.
ax ‐ 531 := 8, where
constant, passes
where a 15a
is a constant,
passes
3
-1y
points (1,
4) and
points
(1,4)
and (x,
(x, 10).
10). What
What is the
the value
value of x?
x?
A)
A) 4
through
point (2,6).
What is the
through the
the point
(2, 6). What
the
x-coordinate of the
x-coordinate
the x-intercept
x-intercept of the
the line?
line?
B) 6
C)
C) 8
D)
D ) 1100
D
Day
ay
Monday
Monday
Thursday
Thursday ”‑
Friday
Friday
Saturday
Saturday
Average speed,
speed , 5
s
(miles per
per hour)
hour)
Number of
Number
calories burned,
burned , c
7.2
616
6.8
584
7.9
672
8.5
720
a
y = bx + c
d
+cC
y = ‐- xX +
y
e
The
lines in the
The equations
equations of two
two perpendicular
perpendicular lines
the
xy-plane are
shown above,
and e
a, b, c, d, and
e
xy-plane
are shown
above, where
where a,b,c,d,
On certain
certain days
week, Elaine
an
days of the
the week,
Elaine runs
r u n s for an
hour on
treadmill. For each
day that
that she
she ran
ran in
each day
hour
on aa treadmill.
the
week, the
the table
shows the
the average
the last
last week,
table above
above shows
average
speeds
at which
speed sat
which she
she ran,
ran, in miles
miles per
per hour,
hour, and
and
the number
number of calories
she burned
burned during
during the
the
the
calories c she
run.. If
relationship between
between cc ands
and s can
can be
be
run
If the
the relationship
modeled
function, which
the
modeled by aa linear
linear function,
which of the
following
functions best
best models
models the
the
following functions
relationship?
relationship?
a
following must
be true?
following
must be
true?
d
A) - < - 1
A)‘1<‐1
e
8
d
B)_1<fl<0
L’
e
B) - 1 < - < 0
A) c(s)
c(s) =
z305+400
A)
30s + 400
d
C)0<§<1
C) 0 <-<
l
e
B ) c(s)
c(s) = 660s
0 $ + 2210
10
B)
d
D)5>1
D) - > 1
8
e
C)
c(s) =
2805+40
C) c(s)
80s + 40
D ) c(s) =
2 990s
0 5-‐ 330
0
D)
If m
m and
and b are
are real
real numbers
and m
m > 0 and
and b > 0,
0,
If
numbers and
then
the
line
whose
equation
is
y
=
mx
+
b
then the line whose equation
= mx +
cannot contain
the following
cannot
contain which
which of the
following points?
points?
A) (0,1)
A)
(0, 1)
B)
B)
C
C))
D)
D)
.
are
If O
0< 2
are constants.
constants. If
b < 1, which
which of the
the
(1,1)
(1,1)
((-‐ 11,1)
,1)
(O,‐1)
(0, - 1)
125
15
Linear
Interpreting
Interpreting Linear
Models
Models
chapter
previous chapter
the previous
extension of the
direct extension
are a direct
On the
SAT, you
you will
w i l l encounter
encounter linear
linear model
model questions
questions that
that are
the SAT,
context,
world
real
a
within
models
these models within a real world context,
the numbers
about
lines. You’ll
interpret the
the meaning
meaning of the
numbers in these
have to interpret
You'll have
about lines.
and y-intercept
slope and
applying
your understanding
y‐intercept to do
do so.
understanding of slope
applying your
equation
the equation
esmn~ted by the
be estimated
can be
EXAMPLE 1: The value
value V,
V, in dollars,
dollars, of aa home
home from 2006 to 2015 can
EXAMPLE
2006.
since
V=
000 ‐- 5, 000T,
where Tis
T is the number
number of years
years
OOOT,where
240,000
= 240,
the meaning
best describes
PART
following best
describes the
meaning of the
the number
number 240,000 in the
the equation?
equation?
Which of the following
PART 1: Whichof
A) The value
value of the home
home in 2006
home in 2015
value of the home
B) The value
home from
C)
value of the
the home
from 2006 to 2015
average value
The average
C} The
increase in the
the value
value of the
the home
home from 2006 to 2015
D) The increase
equation?
the equation?
number 5,000 in the
describes the
following best
best describes
the meaning
meaning of the number
PART 2: Which of the following
A)
B)
C)
D)
The number
number of homes
homes sold
each year
year
sold each
the value
The yearly
yearly decrease
value of the home
home
decrease in the
and in 2015
The difference
between the
value of the
the home
home in 2006 and
the value
difference between
value of the
the home
home per
per square
square foot
the value
decrease in the
yearly decrease
The yearly
-intercept b will
Theyy-intercept
form. The
equation in y =
will
Part 1 Solution:
i l l give
give you
you an
an equation
: mx
mx +
+ b form.
will
que stions w
these questions
Many of these
Solution: Many
and it describe
240,000
they-intercept
case, the
0. In this case,
when x =
typically
an initial
initial value,
value, the value
value when
: O.In
y-intercept is 240,000 and
describess
designate an
typically designate
Answer ~-(A) .
course, is 2006. Answer
which, of course,
after 2006,
the
value of the
home when
when T =
: 0, zero
years after
2006, which,
zero years
the home
the value
designate s
The slope
+ b. The
mx +
form y := mx
Part 2 Solution:
dealing with
w i t h an
an equation
equation of the
the form
slope m
m always
always designates
we're dealing
Again, we’re
Solution: Again,
the
means the
which
000,
5,
is
slope
the slope
case, the
each increase
a rate,
increase in x. In this case,
‐ 5 , 000, which means
decrease in y for each
increase or decrease
the increase
rate, the
Answer ~(B) -.
value of the
decreases by 5,000 for each
each year
year that
goes by. Answer
that goes
home decreases
the home
value
126
THE
THE COLLEGE
COLLEGE PANDA
PANDA
It’s important
important that
you don
don’t
get tricked
into choosing
choosing a
looks right
right but
ultimately
doesn't fit the
the
It's
that you
' t get
tricked into
a rate
rate that
that looks
but ultimatel
y doesn't
context set
set by the
variables xx and
and y (in
( i n this
this case,
case, T and
and V). Answer
Answer ((A)
A ) is wrong
wrong because
we’re n
o t dealing
dealing
because we're
not
context
the variables
with the
number of homes
homes sold;
sold; we're
we’re dealing
dealing with
with the
value of a
Answer (D) is wrong
the
with
the number
the value
a home.
home. Answer
wrong because
because the
numbers in the
the equation
equation aren
aren’t
on aa per-square-foot
per-square‐foot basis.
basis. Always
variables you're
you’re working
numbers
't on
Always be
be aware
aware of the
the variables
working
with.
with.
EXAMPLE 2:
2- The
The maximum
maximum height
height of aa plant
plant h, in inches,
inches, can
by the
EXAMPLE
can bedetermined
be determined by
the equation
equation h =
grow the
the plant.
plant .
where
where x .is
is the
the amount
amount of fertilizer,
fertilizer, in grams,
grams, used
used to
to grow
x:
4
6
“5+
6,,
PART 1: According
According to
to the
the equation,
one more
more gram
gram of fertilizer
fertilizer would
would increase
increase the
the maximum
maximum height
height of a
PART
equation, one
a
plant
plant by how
how many
many inches?
inches?
PART
exactly one
one inch,
how many
PART 2: To raise
raise the
the maximum
maximum height
height of aa plant
plant by
by exactly
inch, how
many more
more grams
grams of fertilizer
fertilizer
should
the plant?
should be
be used
used in growing
growing the
plant?
Part 1 Solution:
Solution: This question
question is essentially
the change
change in hh for every
unit increase
increase in x. This
This is
essentially asking
asking for the
every 11 unit
.
. 4
.
even clearer,
clearer, we can
put the
the
the slope.
slope. From
From the equation,
equation, we can
can see
see that
that the
the slope
slope is
IS ~,
5’ or I-.0.8 I. To make
make this even
can put
the
t
~x +
+ 2. Note
Note that
dealing with
equation into
into y =
equation
: mx
mx +
+ bb form
form by
by splitting
splitting up
up the
the fraction:
fraction: hh =
: Ex
that when
when we're
we’re dealing
with
changes in x and
they-intercept
irrelevant because
a constant
constant that’s
that's always
changes
and y, the
y‐intercept b is irrelevant
because it's
it’s a
always there.
there.
Part 2 Solution:
change in x for every
every l1 unit
unit increase
the reverse
Solution: Because
Because this
this question
question is
is asking
asking for the
the change
increase in h, the
reverse of
Part
Part 1, we need
need to rearrange
rearrange the equation
equation so
so that
that we
we have
have x in terms
terms of h.
4x+6 =2hh
s5
4x + 6
4 x+
+ 66 ==5Shh
4x
4 x=
=5
4x
Shh-‐ 6
S
3
sz-lh‐§
= -h- 4
2
X
Now
that x increases
N o w we can
can see
see that
increases by ~,
Z, or [ill
when
the slope
slope of oour
when h increases
increases by 1. The
The answer
answer is just
just the
u r new
new
equation.
the reciprocal
reciprocal of the
the slope
the original
original equation.
equation. The
equation. A shortcut
shortcut for this
this type
type of question
question is to take
take the
slope of the
The
4 . S
recrproca
' proca 1
l of 4
5 is
5.
reci
1s 4.
5 4
EXAMPLEB:
EXAMPLE
3:
5 -‐ 66m
m
T == 665
A can
the soda,
Fahrenheit, can
can be
can of soda
soda is put
put into
into aa freezer.
freezer. The
The temperature
temperature T of the
soda, in degrees
degrees Fahrenheit,
be found
found by
by
using
equation above,
the can
can has
the freezer.
using the
the equation
above, where
where m is
is the
the number
number of minutes
minutes the
has been
been in the
freezer. What
What is the
the
decrease
the temperature
every 5 minutes
can is left
decrease in the
temperature of the
the soda,
soda, in degrees
degrees Fahrenheit,
Fahrenheit, for every
minutes the
the can
left in the
the
freezer?
freezer?
The slope
represents the
every 1 minute
the can
can is left in the
the freezer. So
slope of -‐66 represents
the change
change in the
the temperature
temperature for every
minute the
So
S x 6 = I30 Idegrees
degrees Fahrenheit.
Fahrenheit.
for every
every 5 minutes,
minutes, the
the temperature
temperature of the
the soda
soda decrease
decreasess by 5
127
CHAPTER
CHAPTER 15
15 INTERPRETING
INTERPRETLNG LINEAR
LINEAR MODELS
MODELS
CHAPTER
page 309.
CHAPTER EXERCISE:
EXERCISE: Answers
Answers for this
this chapter
chapter start
start on
on page
A calculator should NOT be used on the
following questions.
A membership
website offers
offers video
video tutorials
on
membership website
tutorials on
The
large aquarium
aquarium
The water
water level
level h, in feet,
feet, in aa large
h
=
100
3d,
where
can
be
modeled
by
can be modeled by h z 100 T 3d' where d
d is
is the
the
number
of
days
that
have
passed
since
the
number of days that have passed since the
aquarium
last refilled
a q u a r i u m was
was last
refilled.. Based
Based on
on the
the model,
model,
how
level change
each day?
how does
does the
the water
water level
change each
day?
i
A)
Decreases by
A) Decreases
by 3
3 feet
feet
feet
B) Increases
by 3
B)
Increases by
3 feet
A)
every one
one additional
additional video,
video, the
A) For
For every
the site
site
gains
e w members.
500 nnew
members .
gains 500
B) The
available
videos available
The site
site initially
initially made
made 200 videos
members .
to members.
C) Decreases
Decreases by 100 feet
D) Increase
Increasess by 100 feet
feet
‘
programming. The
number of members,
members, m,
programming.
The number
m,
subscribed to the
the site
can be
be estimated
the
subscribed
site can
estimated by the
m = 500 + 200n,
200n, where
the number
equation m
equation
where nn is the
number
videos available
available on
on the
the site.
site . Based
on the
the
of videos
Based on
equation, which
which of the
the following
following statements
equation,
statements is
true?
true?
'
'
The
bread b remaining
The number
number of loaves
loaves of bread
remaining in aa
bakery each
be estimated
bakery
each day
day can
can be
estimated by the
the
the number
number
equation
equation b = 200 -‐ 18h,
18h, where
where h is the
have passed
pass ed since
of hours
hours that
that have
since the
the store's
store’s
opening
the value
in
opening.. What
What is
is the
the meaning
meaning of the
value 18
18in
this
this equation?
equation?
C) The
The site
site was
was able
get 500 members
able to get
members
without any
any available
available videos
without
videos..
D) The site gains 500 new members for every
D) T
h eadditional
S i ’ e g a mvideos
S m Wavailable
membe’sm’eve’y
additional
available
on the
the site.
200
videos
on
site.
s=
10 -‐ 2h
5 = 10
211
recipe suggests
with
A recipe
suggests sweetening
sweetening honey
honey tea with
sugar. The
The equation
equation above
above can
be used
used to
sugar.
can be
to
determine the
the amount
amount of sugar
in teaspoon
s,
determine
sugar s,
5,in
teaspoons,
Ii
that should
be added
added to a
a tea beverage
that
should be
beverage with
with h
teaspoons Of
of honey.
honey. What
What is
the meaning
of the
the 22
teaspoons
is the
meaning of
in
the
equation?
in the equation?
A) The
bread in 18
The bakery
bakery sells all
all its
its loaves
loaves of bread
18
hours.
hour
s.
B)
The bakery
bakery sells
18 loave
loavess of
of bread
bread each
each
B) The
sells 18
hour.
hour.
A)
every teaspoon
teaspoon of
of honey
honey in
in the
the
A) For
For every
beverage, two
more teaspoons
teaspoons of
of sugar
sugar
beverage,
two more
should be added .
ShOUId be added.
B) For
every teaspoon
teaspoon of
of honey
honey in
in the
the
B)
For every
beverage, two
fewer teaspoons
teaspoons of sugar
beverage,
two fewer
sugar
should be
be added.
should
added.
loave s of bread
bread
C) The
The bakery
bakery sells
sells aa total
total of 18
18 loaves
each day
each
day..
D) There
There are
are 18
18 loave
loavess of bread
bread left
left in the
the
bakery at
the end
bakery
at the
end of each
each day
day..
C) For
For every
every two
honey in the
the
two teaspoons
teaspoons of honey
beverage,
more teaspoon
beverage, one
one more
teaspoon of sugar
sugar
should
should be
be added.
added.
For every
every ttwo
honey in the
the
D) For
w o teaspoons
teaspoons of honey
beverage,
one fewer
sugar
beverage, one
fewer teaspoon
teaspoon of sugar
should
added.
should be
be added.
128
THE
PANDA
COLLEGE PANDA
THE COLLEGE
h ==1100
0 0 -~ 44tt
The monthly salary of a salesperson at a used car
dealership is determined by the expression
1,000 + 2, 0OOxc, where x is the salesperson's
commission rate and c is the number of cars sold
by the salesperson. Which of the following
statements is the best interpretation of the
number 2,000 in the context of this problem?
the
model the
used to model
be used
can be
above can
equation above
The equation
The
at
held at
milk held
gallon of milk
a gallon
until a
hours h until
number of hours
number
sour.
goes sour.
Celsius, goes
degrees Celsius,
oft,t, in degrees
temperature of
a temperature
a
following is
the following
which of the
model, which
the model,
on the
Based on
Based
the
number
the
of
interpretation
best interpretation
the best
the
the number 4 in the
equation?
equation?
car at
used car
a used
price of a
average price
The average
A) The
at the
the
B)
C)
D)
gallon of
make a gallon
will make
°C will
increase of 11°C
A) An increase
faster.
sour 4 hours
go sour
milk go
milk
hours faster.
dealership
dealership
at
salesperson at
a salesperson
salary of a
monthly salary
base monthly
The
The base
dealership
the dealership
the
eamed
commission earned
monthly commission
average monthly
The
The average
dealership
the
at
salesperson
each salesperson the dealership
by each
sold by the
cars sold
number of cars
average number
The
The average
the
month
each month
dealership each
dealership
gallons of
make 4 gallons
will make
°C will
increase of 1
B) An increase
1°C
faster .
hour faster.
sour 1 hour
milk go sour
milk
gallon of
make a gallon
will make
increase of 4°C will
C) An increase
faster.
hour faster.
sour 1 hour
go sour
milk go
milk
gallon of
make a gallon
will make
4° C will
increase of 4°C
D) An increase
faster.
hours faster.
sour 4 hours
go sour
milk go
milk
p = 2, OO0s+ 15, 000
p=2,0005+15,000
an auction
at an
was sold
lamp was
antique lamp
An antique
sold at
auction.. The
The
the
during the
dollars, during
lamp, in dollars,
the lamp,
price p of the
price
equation
the equation
by the
modeled by
be modeled
can be
auction can
auction
seconds
number of seconds
the number
where t is the
lOt, where
p = 900 ‐- 10t,
what
model,
the
to
According
auction.
the
in
left
left the auction. According to the model, what
equation?
the equation?
the 900 in the
meaning of the
the meaning
is the
above to
equation above
the equation
uses the
government uses
state government
A state
town
a town
population p for a
average population
the average
estimate the
estimate
best
following best
the following
Which of the
schools. Which
withs 5 schools.
with
the
number 2,000 in the
the number
meaning of the
the meaning
describes
describes the
equafion?
equation?
lamp
the lamp
price of the
auction price
starting auction
The starting
A) The
A)
lamp
the
of
price
auction
final
The
B)
B) The final auction price the lamp
per
lamp per
the lamp
price of the
the price
increa se in the
The increase
C)
C) The
second
second
lamp, in
the lamp,
auction off the
took to auction
time it took
The time
D) The
seconds
seconds
students at
number of students
average number
The average
A) The
at each
each
a ttown
school in a
school
own
each
schools in each
number of schools
average number
The average
B) The
ttown
own
town 's
a town’s
increase in a
estimated increase
The estimated
C) The
C)
additional
each
for
population
population
each additional school
school
a town
population of a
estimated population
The estimated
D) The
D)
town
schools
without any
without
any schools
y=
1.30x‐
- 1.50
= l.30x
above to
equation above
the equation
uses the
teller uses
bank teller
A bank
where y is the
euros, where
into euros,
dollars into
U.S. dollars
exchange US
exchange
the
dollar amount.
U.S. dollar
the US.
and x is the
amount and
euro amount
euro
amount.
interpretation
best interpretation
the best
following is the
the following
Which of the
Which
equation?
the equation?
the 1.50 in the
of the
the
do the
euros to do
charges 1.50 euros
bank charges
The bank
A) The
A
exchange.
currency exchange.
currency
the
do the
dollars to do
U.S.
charges 1.50 U
bank charges
The bank
B) The
S . dollars
B)
exchange.
currency exchange.
currency
euros.
worth 1.50 euros.
dollar is worth
U.S. dollar
One US.
C) One
C)
dollars.
worth 1.50 U.S.
euro is
One euro
D) One
is worth
U S . dollars.
D)
129
CHAPTER
15 INTERPRETING
INTERPRETING LINEAR
MODELS
CHAPTER 15
LINEAR MODELS
calculator is allowed
allowed on
on the following
following
A calculator
questions.
questions.
Which of the following is the best interpretation
of the y-intercep t in the contex t of this problem?
A) The price of each cake
2x + 9
t_2x+9
t=--
B) The cost of making each cake
_ 55
C) The daily costs of running a bakery
The
equation above
models the
the time
time t, in
The equation
above models
seconds, it takes
takes to load
load aa web
web page
with x
seconds,
page with
images. Based
Based on
on the
model, by how
how man
many
images.
the model,
y
seconds
does each
each image
the load
load time
seconds does
image increase
increase the
time
of
a
web
page?
ofawebpage?
D) The daily cost of making the cakes that
weren't ab le to be sold
_
What
mean that
solution to the
the
What does
does it mean
that (5, O)is
0) is aa solution
equation
equation of the
the line?
line?
A) The
needs to sell 55 cakes
to
The bakery
bakery needs
cakes per
per day
day to
cover
cover its daily
daily expenses.
expenses .
B) Each
must be
be sold
sold for at
Each cake
cake must
at least
least 5 dollars
dollars
to cover
t,
cover the
the cost
cost of making
making iit.
C)
cake.
C) It
It costs
costs 55 dollars
dollars to
to make
make each
each cake.
D) Each
day, the
Each day,
the bakery
bakery gives
gives the
the first 5 cakes
cakes
away for free.
away
---------T---------7
Questions 11-13
Questions
11'13 refer
refer to
to the
the following
following
information.
information.
Daily Profit
Profit (y)
Daily
---------
600 -1----,----,-1 ------
500
400 -
L -
300 -l- ---r----"'-------"'-1
200 +- -,----,-----100 +- --,----,------,--
;-;,"'--~-
----;;;,
~·
~~--
1---4----+--+--+~~+--+----t---;,-100
-200
-300 .
-400
A ---------
Cakes
7- - 8--- 9--
(x)
T
= 56 + Sh
To w
a r m up
room, Patrick
the
warm
up his
his room,
Patrick turns
turns on
on the
heater.
his room,
room, in
heater . The
The temperature
temperature T of his
degrees
Fahrenheit,, can
modeled by the
the
degrees Fahrenheit
can be
be modeled
equation
equation above,
above, where
where h is the
the number
number of hours
hour s
since
since the
the heater
heater started
started running.
running . Based
Based on
on the
the
model,
what is the
model, what
the temperature
temperature increase,
increase, in
degrees
Fahrenheit,
30 minutes
degrees Fahrenhei
t, for every
every 30
minutes the
the
heater
on?
heater is turned
turned on?
-500
-600
-700 ·
The relationship
between the
the daily
daily profit
The
relationship between
profit y, in
dollars,, of aa bakery
bakery and
and the
the number
number of cakes
cakes sold
sold
dollars
by the
bakery is graphed
graphed in the
the xy-plane
xy-plane above
above..
the bakery
What does
the slope
slope of the
line represent
represent??
What
doe s the
the line
The price
price of each
cake
A) The
each cake
The profit
from each
cake sold
sold
B) The
profit generated
generated from
each cake
C) The
The dail
daily
profit generated
from all the
C)
y profit
generated from
the
cakes that
that were
were sold
sold
cakes
D) The
The number
number of cakes
cakes that
that need
need to be
be sold
sold to
make aa daily
dollars
make
daily profit
profit of 100 dollars
130
130
THE COLLEGE PANDA
2
4
2yy-‐ xX ==114
add
like to add
would like
but would
frog but
pet frog
a pet
Alice
owns a
Alice owns
veterinarian
local veterinarian
The local
tank . The
same tank.
the same
turtles to the
turtles
the total
determine the
above to determine
equation above
uses
total
the equation
uses the
be
that should
gallons, that
watery,y, in gallons,
amount
should be
amount of water
alongside
thrive alongside
turtles to thrive
tank for x turtles
the tank
held
held in the
the
which of the
equation, which
the equation,
on the
Based on
frog. Based
Alice’s
Alice's frog.
true?
be true?
following must
must be
following
support
water can
gallon of water
1,
can support
additional gallon
One additional
I. One
turtles .
more turtles.
ttwo
w o more
requires two
turtle requires
additional turtle
One additional
III.
I , One
t w o more
more
water.
gallons
gallons of water.
half
an additional
requires an
turtle requires
more turtle
One more
III. One
additional half
water.
a
gallon of water.
a gallon
A)
B)
C)
C)
D)
only
ll only
II
IIII
I I only
only
I and
only
and II only
only
and III
I and
I I I only
C
=
1.5 + 2.5x
equation above
the equation
uses the
post office uses
local post
A local
above to
a
dollars, of mailing
cost C, in dollars,
the cost
determine
mailing a
determine the
increase of 10
pounds . An increase
weighing x pounds.
shipment weighing
shipment
10
equivalent to
mailing cost
the mailing
dollars in the
dollars
cost is equivalent
to an
an
weight of
the weight
pounds in the
many pounds
how many
increase of how
increase
the
shipment?
the shipment?
A) z2
B) 2.5
C) 4
D)
0) 5
131
131
Functions
Functions
function is aa machine
machine that
that takes
takes an
an input,
input, transforms
transforms it,
u t an
functions are
are
A function
it, and
and spits
spits oout
an output.
output. In math,
math, functions
denoted
x ) , with
with x being
being the
input. So
50 for the
the function
function
denoted by Jf ((x),
the input.
f(.\') : x2 +1
every
squared and
and then
o n e to get
get the
the output.
It's important
important to understand
understand that
every input
input is squared
then added
added to one
output. It's
that x is aa
completely
label‐it’s
just aa placeholder
placeholder for the
the input.
input. In fact, II can
put in
in whatever
whatever II want
w a n t as
completely arbitrary
arbitrary label
- it's just
can put
as the
the
input, including
input,
including values
values with
with x in them:
them:
ff<2x>
(2x) =
: (2x)2
(2.102 +
+11
Jf(a):a2'+1
(a) = a 2 + 1
b+
: (b
(17+
1)2+1
Jf ((b
+11)) =
+ 1)2+
1
fW(*)
) =
r (*)2+
( * ) ‘ * 1l
f(Panda)) =
: (Panda
(Panda)2
+11
/(Panda
)2 +
2 Wrap the
Notice the
Notice
the careful
careful use
use of parentheses
parentheses.. In the
the first equation,
equation, for example,
n o t the
the same
same as
2x2.
example, (2.1')2
(2x)2 is not
as 2x
. Wrap the
input
parentheses and
and you'll
you’ll never
never go
go wrong
wrong..
input in parentheses
EXAMPLE 1: If
I f J(x)
f(x) =
: (11+
1)‘, then
then what
what is the value
value off(0)
(1) +
+ f/(2)
( 2 ) + f/(3)?
(3) ?
EXAMPLE
(x + l)X,
of J(O) + ff(l)
Just plug
the inputs.
Just
plug in the
inputs .
fJ((O)
0 )++ Jf ((1)
l ) f+f J( 2
3) =
z(0+1)“+(1+1)1+(2+1)2+(3+1)3
(2)) ++fJ( (3)
(0 + 1)0 + (1 + 1) 1 + (2 + 1)2 + (3 + 1)3
:1“+2‘+32+4~‘
=
1°+ i + i + 43
: 11f+22¢+99++664
4
=
z fi_. f f
=~
132
132
COLLEGE PANDA
THE COLLEGE
PANDA
EXAMPLE
EXAMPLE2:
4
4
f(x)
2 -1 0x+25
f(x) = xx2‐10x+25
For what
the function
function f above
aboveundefined?
For
what value
value of x is the
undefined?
Because we
we can't
can’t divide
divide by 0,
O,a
function is
is undefined
undefined when
the
Because
a function
when the denominator
denominator is zero.
zero. Setting
Setting th
e denominator
denominator
zero,
to zero,
10x +
xx22 -‐ lOx
+ 25 z= 0
(x ‐- 5)2
5)2z= 0o
x
X
=z [fil
The
when x = 5.
5.
The function
function f is undefined
undefined when
This wou
would
be aa good
time to talk
about domain
domain and
and range
range::
This
ld be
good time
talk about
Domain: The
The set
set of
of all
all possible
input values
values (x)
lead to
•0 Domain:
possible input
(x) to
to a
a function
function (values
(values that
that don’t
don 't lead
to an
an invalid
invalid
operation
operation or an
undefined output)
output)..
an undefined
0
Range: The
all possible
output
values (y)
(y) from
• Range:
The set of all
possible ou
tput values
from a function.
function .
In Example
2, x z= 5 leads
However, all other
other val
values
give real
real number
outputs.
Examp le 2,
leads to Jf (x)
(x) being
being undefined.
w1defined. However,
ues of x give
number outputs.
Therefore,
numbers except
except 5.
5. To
To verify,
we can
take aa look
at the
the grap
graph
Therefore, the
the domain
domain of
offf is all real
real numbers
verify, we
can take
look at
h of
off:f:
y
5
0
5
As
the graph
graph has
has no
no y-va
y-value
when x =
= 5.
5. In
In fact,
fact, x =
= 5 is
is like
like an
an invisible
invisible line
graph
As you
you can
can see,
see, the
lue when
line that
that the
the graph
approaches
approaches but
but never
never crosses
crosses.. We
We call these
these lines
vertical asymptotes.
summarize, the
has one
one
lines vertical
asymptotes . To summarize,
the function
function f has
verticall asymptote
with equation
equation x =
= 5.
5.
vertica
asymptote with
You might've
might’ve also
also noticed
noticed that
that the
the graph
graph never
never goes
goes below
below the x‐axis.
approaches
x-ax is. It’s
It's another
ano ther line
line that
that the
the graph
graph approaches
but
never crosses
crosses.. The
x-axis, in this
is aa horizontal
horizontal asymptote.
has one
o n e horizontal
horizontal
but never
The x-axis,
this case,
case, is
asymptote . The
The function
function f has
asymptote with
asymptote
with equation
equation y = 0.
Because
are no
on the
graph that
that have
have a
y‐value of O
Oor
of f is
is all positive
positive
Because there
there are
no points
point s on
the graph
a y-value
or below,
below, the range
range of
real
Put mathematically,
f (x) > 0.
0. By
thiss makes
sense. Because
Because of
of the
the square
square in
in the
the
real numbers.
numbers. Put
mathematically, f(x)
By the
the way,
way, thi
makes sense.
denominator
(x) =
= xm2
denominator of
of ff(x)
_
l~x
+ 25
= fi,
(x
you always
positive output
output for
any va
value
in the
the
~ 5 ) 2 , you
alwa ys get
get a
a positive
for any
lue of
of xx in
domain.
domain .
Let’s
the domain
domain,, start
start with
with all
all real
and then
the values
values of xx for
for which
Let's summarize.
summarize. To find the
real numbers
numbers and
then exclude
exclude the
which
the function
function is invalid
invalid or undefined.
For examp
example,
2 0
take the
the
the
undefined . For
le, the
the domain
domain of y := fl.jx is
is xx ~
0 because
because we
we can't
can't take
square
negative numbers
numbers..
square root
root of negative
To find the
graph the
on your
your calculator
calculator and
u t the
possible values
values of
of y,
y, taking
taking note
note of
of
the range,
range , graph
the function
function on
and figure
figure o
out
the possible
any
asymptotes.
any horizontal
horizontal asymptotes.
133
CHAPTER 16
CHAPTER
16 FUNCTIONS
FUNCTIONS
EXAMPLE
3:I f f ( x -‐ 1)
1) =
and g(x) == xx +
3,what is the value (2))?
off(g(2)) ?
EXAMPLE3:If/(x
= 6x
6xandg(x)
+ 3,whatisthevalueof/(g
Whenever you
of other
other functions),
functions), start
Whenever
you see composite
composite functions
functions (functions
(functions of
star t from
from the
the inside
inside and
and work
work your
your way
way
out.
out. First,
First,
g( 2) = 2 + 3 = 5
Now
we have
have to
(5).
Now we
to figure
figure out
out the
the value
value of
of f/(5).
Well, we
into J(x
f ( x -‐ 1)
1) == 6x
6x to
to get
getf(5)
= 6(6)
= I-.36 I.
we can
can plug
plug in
in x
x := 6
6 into
f (5) =
6(6) =
EXAMPLE
byf(x)
EXAMPLE 4:
4: Functions
Functions/ f and
and g
g are
are defined
defined by
f(x)
the value
value of k ?7
i·
= xx+1
= §.1ff(g(f(k)))
= 10,
what is
is
=
+ 1 and
and g(x)
g(x) =
If f(g(f(k))) =
10, what
Again,
our way
way out:
out:
Again, we
we start
start from
from the
the inside
inside and
and work
work our
f ((k)
k )=: k + 11
= k+1
g(k+1):k%1
2
g(k + 1)
1c;1)
= k ;1+
1
Finally,
Finally,
k+l
-I%1+1210
2- + 1 = 10
k+1 =
k%1:9
9
2
8
kk + 1l 2=118
= [ill
k=E
k
As we've
returns an
and output
output pairs
allow
we 've mentioned,
mentioned, a function
function takes
takes an
an input
input and
and returns
an output.
output. Well,
Well, these
these input
input and
pairs allow
us
function
asa
set
of
points
in
the
xy‐plane,
with
the
input
as
x
and
the
output
asy.
In
us to graph
graph any
any function as a set of points in the xy-plane, with the input as x and the output as y. 1n fact,
fact,
2
y = xx2+
the same
1. Both
and yy are
are the
+ 11is
is the
same asf
as f (x) = x2+
x2 + 1.
Both ff(x)
(x) and
the same
same thing‐they’re
thing-they're used
used to
to denote
denote the
the output.
output.
The
reason we use
with the
The only
only reason
use y is that
that it’s
it's consistent
consistent with
the y‐axis
y-axis being
being the
the y‐axis.
y-axis.
Anytime
(x) is used
of it
it as
as the
(x) >
> 0,
Anytime ff(x)
used in a graphing
graphing question,
question, think
think of
they.y. 50
So ifif a
a question
question states
states that
that fJ(x)
0, all
allyy
values
graph is
is always
above the
x-axis.
It’s
extremely
important
that
you
learn
to
think
values are
are positive
positive and
and the graph
always above
the x-axis. It's extremely important that you learn to think
of points
inputss and
and outputs
points on
on a
a graph
graph as
as the
the input
outputs of
of a
a function.
function .
134
THE
PANDA
THE COLLEGE
COLLEGE PANDA
EXAMPLES:
EXAMPLE
5:
y
;
I/\
\
J
- 1-
'O
I
I
-·
\
\ I
\
X
\
\ I
\
The
f (x) is
is shown
shown in
in the
xy-plane above.
above‘ For
what value
value of
(x) at
The graph
graph of
of J(x)
the xy-plane
For what
of x
xisis fJ(x)
at its
its maximum?
maximum?
Again,
(x) as
as they
the y.. So
So we’re
graph with
with the
Again, when
when it
it comes
comes to
to graphs,
graphs, think
think of
offf (x)
we're looking
looking for
for the
the point
point on
on the
the graph
the
highest
y‐value, the
”peak”" of
That point
point is
x-value
is
..
highest y-value,
the "peak
of the
the graph.
graph. That
is (5,
(5, 4),
4), and
and so
so the
the x-value is I}].
EXAMPLE
the function
function with
with equation
= ax
a):22 +
point (1,
2), what
what is
EXAMPLE6: If the
equation yy =
+ 33 crosses
crosses the
the point
(1,2),
is the
the value
value of
of a
a??
Remember - a point
Remember‐a
is just
an output,
output, an
an xx and
y. Because
2) is
is a
on the
the graph
point is
just an
an input
input and
and an
and a
a y.
Because (1,
(1, 2)
a point
point on
graph of
of the
the
function,
function, we
we can
can plug
plug in 1 for x and
and 22 for y.
2
2 =:a(1)2
a(1 )2++3
3
2 = aa + 3
a z j
2+
EXAMPLE
EXAMPLE 7: If the
function y =
= xx2
+ 2x -‐ 4 contains
the point
and m
m>
what is
is the
value
the function
contains the
point (m,2m)
(m, 2m) and
> 0,
0, what
the value
of
m?
ofm?
It's important
important n
not
get intimidated
It's
o t to get
by all
the variables.
variables. The
gives us
us aa point
point on
on the
graph, so
so let's
let's
intimidated by
all the
The question
question gives
the graph,
plug
plug it
it in.
in .
: x2
yy =
x2 +
+2x
2x -‐ 4
4
2m = m2
+ 2m
2m-‐ 44
2m
m2 +
00 = m2
m 2 -‐ 4
4
From here
From
we can
can see that
:l:2.
The question
question states
states that
: l.@].
here we
that m = ±
2. The
that m > 0,
0, so
so m
m=
The zeros,
zeros, roots, and x-intercepts
The
function are
are all
for the
the same
thing‐the
values
x-inter~epts of aa function
all just
just different
different terms
terms for
same thing
- the values
of x that
that make
make ff(x)
(x) =
= 00 (or y =
= 0).
O). Graphically,
Graphically, they
values of
crosses
they refer
refer to
to the
the values
of x
x where
where the
the function
function crosses
the x-axis.
the
135
FUNCTIONS
16 FUNCTIONS
CHAPTER 16
CHAPTER
EXAMPLES:
EXAMPLE8:
y
~
I
I
I
,
--
I
I
I
I
I
'
.
""
.."
.,_:.
I
I
'\
--
'\
I
I
'\
'
X
'
I
I
I
I
" ...../
I
I
The graph off (x)
= x3 - 2x2- Sx + 6 is shown in the xy-plane
above.
does the
zeros does
distinct zeros
PART 1: How
How many
many distinct
the function
function ff have?
have?
value
the value
be the
could be
following could
the following
which of the
solution, which
that J(x)
such that
constant such
If k is a constant
PART 2: If
f (x) =
= k has
has 1 solution,
of kk??
-3
A) ‐3
A)
B) 1
C) 5
0)99
D)
has
so ff has
times, so
three times,
x-axis three
the x-axis
crosses the
graph crosses
The graph
Solution : The
Part 1 Solution:
Part
and 3.
are -‐ 22,, 1, and
zeros are
these zeros
that these
see that
see
can
we can
the graph,
distinct zeros
ffidistinct
zeros.. From
From the
graph, we
Read
explanation. Read
the explanation.
during the
lost during
you feel lost
panic ifif you
so don't
involved, so
quite involved,
question is quite
This question
So lution: This
Part 2 Solution:
don’t panic
sense
make
to
able
be
you'll
promise
I
.
confusing
were
that
bits
the
to
back
go
then go back the bits that were confusing. promise you'll be able make sense of
and then
through and
the way
all the
all
way through
everything.
everything.
always
we always
input, we
the input,
matter the
function. No matter
just a function.
realize that
this question,
To truly understand
question, first realize
that a constant
constant is just
understand this
does
What does
say k = -‐ 33.. What
let's say
So let’s
y = k or g(x) = k. So
can write
we can
question, we
this question,
output. In this
same output.
the same
get the
get
write it as
asy
at -‐ 33!!
line at
horizontal line
like? A horizontal
y z= ‐3
look like?
- 3 look
y
intersection points
the intersection
referring to the
merely referring
it's merely
= k, it’s
so lutions to fJ (x) =
the solutions
asks for the
question asks
a question
Now
points of f (x)
when a
Now when
(x)
J
other,
each other, f(x) =
equal to each
functions equal
sets ttwo
question sets
a question
general , if a
k. In general,
line y =
horizontal line
and
= k.
w o functions
= g(x), and
and
the horizontal
and the
intersection points
the intersection
only at the
it's only
all, it’s
After all,
points. After
intersection points.
the intersection
referring to the
it's referring
solutions, it’s
the solutions,
about the
asks
points
you about
asks you
constant
a constant
be a
happens to be
just happens
g(x) just
case, g(x)
particular case,
this particular
functions. In this
both functions.
that
same for both
the same
value of y is the
the value
that the
function,
: k.
g(x ) =
function , g(x)
there
above, there
shown above,
as shown
- 3 as
So ifif k =
points. 50
intersection points.
the number
The
equivalent to the
number of intersection
: ‐3
solutions is equivalent
number of solutions
The number
the
are the
themselves are
solutions themselves
The solutions
points. The
must
intersection points.
the 3 intersection
represented by the
as represented
solutions to fJ (x) = -‐ 33,, as
be 3 solutions
must be
and 2.6.
2.2, 1.6, and
be -‐2.2,
them to be
x-values
estimate them
can estimate
points . We can
those points.
x-values of those
136
THE COLLEGE
COLLEGE PANDA
THE
PANDA
Getting back
we have
choose a
Getting
back to the
the original
original problem,
problem, we
have to
to choose
ak
k such
such that
that there
there is
is only
only one
one solution.
solution. Now
Now we’re
we're
thinking backwards.
thinking
Instead of being
constant, we
have to
to choose
choose it.
it. Where
might
we place
place aa
backwards. Instead
being given
given the
the constant,
we have
Where might we
horizontal
there’s only
only one
intersection point?
o t at
we just
just showed
how
horizontal line
line so
so that
that there's
one intersection
point? Certainly
Certainly nnot
at ‐3
- 3 because
because we
showed how
that would
would result
that
result in 3 solutions.
solutions .
Well, looking
looking back
one just
below -‐ 44.. Horizontal
Horizontal lines
at these
back at
at the
the graph,
graph, we
we could
could place
place one
just above
above 88 or
or just
just below
lines at
these
values
w i t h fJ (x) just
just once.
Looking at
is the
values would
would intersect
intersect with
once. Looking
at the
the answer
answer choices,
choices, 9
9 is
the only
only one
one that
that meets
meets our
our
condition. Answer
Answer ~(D) .condition.
Let’s
take a
revisit part
part 1.
1. In
In part
1, we
we found
number of
intersection points
between fJ (x)
and
Let' stake
a moment
moment to revisit
part 1,
found the
the number
of intersection
points between
(x) and
But realize
realize that
that the
just the
the horizontal
horizontal line
y
=
0.
In
counting
the
number
of
intersection
the
the x-axis.
x-axis . But
the x-axis is
is just
line y = 0. In counting the number of intersection
points
= 0,
really doing
doing is
is finding
the number
number of
points between
between f (x) and
and the
the horizontal
horizontal line
line yy =
0, what
what you
you were
were really
finding the
of
solutions
0.
solutions to f (x) =
=
If you
you didn’t
in this
this example
the first
first time
through
didn't grasp
grasp everything
everything in
example the
time through,
through, it’s
it's ok.
ok. Take
Take your
your time
time and
and go
go through
it again,
making sure
throw quite
questions at
at
again, making
sure you
you fully
fully understand
understand each
each of
of the
the concepts.
concepts . The
The SAT
SAT will
will throw
quite aa few
few questions
you
as well
well as
as the
the solutions
solutions to
(x) =
= g(x).
you related
related to the
the zeros
zeros of
of functions
functions as
to fJ (x)
g(x) .
Hopefully by
by now,
you’re
starting
to
see
constants
ashorizontal
lines.
5, that
means
Hopefully
now , you're starting to see constants as horizontal lines. 50
So for
for instance,
instance, if
if fJ (x)
(x) >
> 5,
that means
the
is above
z 5.
in this
i l l help
help you
you on
on a
the entire
entire graph
graph of
offf is
above the
the horizontal
horizontal line
line y =
5. Thinking
Thinking of
of constants
constants in
this way
way w
will
a
lot of SAT graph
lot
graph questions.
question s.
3 + 2x2
EXAMPLE 9:
Which of
following could
could be
be the
the graph
EXAMPLE
9: Which
of the
the following
graph of
of y
y = xx3+
2x 2 +
+ xx ++ 11??
B)
m
A)
~
C)
q
#‘x $3: %x
yy
y
y
D)
yy
x
Although
Although the
the given
function looks
complicated and
you might
given function
looks complicated
and you
might be
be tempted
tempted to
to graph
graph itit on
on your
your calculator,
calculator, this
this
is the
find a
be on
on the
graph
and eliminate
eliminate
the easiest
easiest question
que stion ever!
ever! A
Alll l you
you have
have to
to do
do is
is find
a point
point that’s
that's certain
certain to
to be
the graph and
the
So what’s
an easy
easy point
and test?
test?
the graphs
graphs that
that don’t
don 't have
have that
that point.
point. So
what 's an
point to
to find
find and
Plug
to get
= 1.
1. N
o w which
which graphs
graphs contain
contain the
Plug in x
x z
= 00 to
get y =
Now
the coordinate
coordinate (0,
(0, 1)?
1)? Only
Only graph
graph ~-.
-
By
are particularly
particularly good
good for
to use
By the
the way,
way, numbers
numbers like
like 00 and
and 11 are
for finding
finding ”easy”
"easy" points
points to
use for
for this
this strategy.
strategy.
137
CHAPTER 16
CHAPTER
16 FUNCTIONS
FUNCTIONS
Function Transform
Function
Transformations
ations
Function transformations
Function
transformations are
are changes
changes we
we can make
make to
to the
the equation
equation of
of a
a function
function to
to ”transform”
"transform" its
its graph
graph in
in
specific ways.
The
transformations
you
might
encounter
on
the
SAT
are
reflection
across
the
x-axis,
ways .
transformations you might encounter on the SAT are reflection across the x-axis, vertical
verticaJ
shift, horizontal
shift, and
We’ll cover
first three
shift,
horizontal shift,
and absolute
absolute value,
value. We'll
cover the
the first
three here
here and
and discuss
discuss absolute
absolute value
value transformations
transformations
in the absolute
absolute value
value chapter.
chapter.
Let's start
Let’s
with the
(x) =
z x2
+ 2x,
start with
the example
example function
function fJ(x)
x2 +
2x, whose
whose graph
graph looks
looks like
like
y
graph of fJ(x)
To reflect the graph
(x) across
down),, multiply
( x ) by
by -‐ 11.. The
The resulting
across the x-axis
x-ax is (flip
(flip itit upside
upside down)
multiply ff(x)
resulting equation,
equation,
2 - 2x, produces
y
=
f(x)
=
x
the
reflected
312 ~f (x) z ‐x2 ‐
produces
reflected graph.
graph.
yy =
= -‐ f(x)
fl fl
y
·. I
To shift
shift the graph
up, add
To
graph of fJ(x)
( x ) up,
add a constant
( x ) . For example,
( x ) ++ 22 =z x2
+ 2x
constant to ff(x).
example, y 2= fJ(x)
x2 +
2x +
+ 22 produces
produces aa
graph that
that is 2 units
graph
(x).
units above
above the graph
graph of
of ff(x).
shift the graph
graph of fJ(x)
down, subtract
To shift
(x) down,
(x). For
(x) -‐ 2
+ 2x
subtract a constant
constant from ff(x).
For example,
example, yy =
= fJ(x)
2 z= x2
x2 +
2x -‐ 2
2
produce s a
a graph
produces
graph
of
f
(
x).
graph that
that is 2 units
units below
below the
the graph of J(x).
yy =
= f(x
fl n) ++22
y = f(x) - 2
y
y
138
COLLEGE PANDA
THE COLLEGE
PANDA
To shift
x ) to the
the left by a
[1units,
units, substitute
substitute x +
+ a in
= fJ((xx +
+ 1)
1) =
=
shift the
the graph
graph of fJ((x
in for
for x.
x. For
For example,
example, yy =
1)22 + 2(x + 1)
1) produces
produces a
graph that
that is
unit to
left
of
the
graph
of
f(x).
(x + 1)
a graph
is 11 unit
to the
the left of the graph of J (x).
To shift
x ) to the
substitute x -‐ a
shift the
the graph
graph o
offff ((x)
the right
right by a11units,
units, substitute
a in
in for x.
x . For
For example,
example, yy =
= fJ((xx -‐ 1) =z
+ 2(x -‐ 1)
1) produces
is 11 uunit
n i t to
off(x).
(x -‐ 11)2
)2 +
produces a
a graph
graph that
that is
to the
the right
right of
of the
the graph
graph of
J (x).
y = f (x + l )
y
= J (x -
y
1)
y
Here's
Here’s a trick that
use to make
make sense
that I like to use
sense of horizontal
horizontal shifts:
shifts :
For horizontal
For
horizontal shifts,
shifts, find
find out
out what
what value
value of
of xx makes
makes the
equal to
the substituted
substituted expression
expression equal
to 0.
0. This
This value
value will
will
tell
tell you
horizontal shift
shift is.
is. For
For instance,
instance, when
when we
we have
f
(x
‐
1),
what
value
of
x
makes
1
you what
what the
the horizontal
have J(x - 1), what value of x makes x
x -‐ 1
equalto
0?x
=
1.
Sof(x
‐
1)is1
unit
tothe
rightoff(x).
What
aboutf(x+4)?
Well,x
=
‐4makesx+4
equal to 0? x = 1. So J(x - 1) is 1 unit to the right off (x ). What about J(x + 4)? Well, x = - 4 makes x + 4
equal to
equal
to 0. Since
value is negative,
graph is
is shifted
left. And
And what
what about
f(3x -‐ 2)?
Since the
the value
negative, the
the graph
shifted 4 units
units to
to the
the left.
about /(3x
2)?
What's5 the
the horizontal
horizontal shift
What’
relation to
to f(x)?
f (x)? A value
value of x =
= ~
2 makes
equal to
the horizontal
horizontal
shift in relation
makes 3x -‐ 22 equal
to 0,so
0, so the
2
shift is
is ~3 units
units to
right.
shift
to the
the right.
Note
that horizontal
horizontal and
Note that
referred to astranslations.
the graph
of a
and vertical
vertical shifts
shifts are
are commonly
commonly referred
as translations. And
And the
graph of
a transformed
transformed
hmction is often
function
often called
called an
an image
image of the
graph of the
the original
original function.
the graph
function .
EXAMPLE
EXAMPLE 10: In the
the function
function g
units to
to the
the left
left
the xy‐plane,
xy-plane, the
the graph
graph of the
g is the
the graph
graph of
of f/ translated
translated 55 units
and 33 units
units downward.
downward. If
and
the function
(x) = (x -‐ 3)
3)22 +
+ 1,
1, which
which of
of the
the following
lithe
function f is
is defined
defined by fJ(x)
following defines
defines
g(x)?
g(x)?
A)g(x)
A)g(x)=(x‐8)2‐2
= (x - 8)2 - 2
B)g(x)=(x+2)2‐2
B)g(x) = (x + 2)2 - 2
C)g(x)=(x‐8)2+4
C)g(x) = (x - 8)2 + 4
D)g(x)=(x+2)2+4
D)g(x) = (x+2) 2 +4
A translation
units downward
downward means
translation of 5 units
units to the
the left and
and 3 units
means that
that
g(
J (x + 5) -‐33
3 0x )) =
=f(-Y+5)
: ( (x
( x + 55)) -‐ 33)) 22+
+ 11-_ 33 == (x+2)2~2
=
(x + 2) 2 - 2
Answer
Answer ~(B) .
To summarize,
summarize, for a function
f (x),,
To
function J(x)
o
(x)) results
in a
x-axis
• -‐ Jf (x
results in
a reflection
reflection across
across the
the x-axis
0
x) +
+ a results
units; Jf(.r)
of a
a units
units
• fJ((x
results in
in an
an upward
upward shift
shift of a units;
(x) ‐- 11
a results
results in
in a
a downward
downward shift
shift of
0
+ a) results
shift of a units
horizontal shift
shift of
a units
• f (x +
results in a horizontal
horizontal shift
units to
to the
the left;
left; f (x ‐- (1)
a) results
results in
in a
a horizontal
of a
w1its to
to
the
right
the right
139
CHAPTER
CHAPTER 16
16 FUNCTIONS
FUNCTIONS
CHAPTEREXERCISE:Answers for this chapter start on page 311.
A calculator should NOT be used on the
following questions.
y
X
y
0
20
1
21
3
29
The table
table above
above displays
displays severa
The
severall points
points on
on the
the
graph of the
the function
function fin
graph
f in the
the xy-plane.
xy-plane. Which
Which of
the
following could
could bef
be J(x)?
the following
(x) ?
A)
f (x)) = 20x
A) J(x
B) fJ(x
(x)) z= xx ++20
8)
20
C) fj (x)
(X) =
= xX -‐ 20
20
C)
2
D)
D) f(x)
J(x )==x2
x ++20
20
The graph
graph of the
the function
function f is shown
the
The
shown in the
xy-plane
above.
If
J(a)
=
f(3),
which
xy-plane above. If f (a) f (3), which of the
the
following could
could be
be the
following
the value
value of
of aa ??
A)) -‐ 4
A
8) ) -‐ 3
B
C)) -‐ 2
C
D) 1
y
J(x)
y
In the portion of the xy-plane shown above, for
how many values of x does J(x ) = g(x)?
A) None
B) One
The function f is graphed in the xy-plane above .
For how many values of x does J(x) = 3?
C) Two
D) Three
A) Two
8) Three
C) Four
D) Five
140
THE COLLEG
COLLEGE
THE
E PANDA
PANDA
For which of the following functions is it true
that
J(-3) ==f(3)
J(3)??
thatf(‐3)
IfIf fJ(x)
(x) =
values
= x2,
x 2 , for
for which
which of
of the
the following
following va
lues of
of
c isf
is ((c)
c) < cc??
A) f
J(x)
M
m ==-X2;
1
A) 1
A)E
2
x3
B
8) ) 1
8) J(x) =
B)fu%=§
3
3
C) ~
C)z
2
C)
C) f(x)
J(x) =3x2+1
= 3x 2 + 1
D) f(x)
0)
f(x) = xx +
+ 22
D) ) 2
D
the graph
graph of the
IfIf the
the function
has x-intercepts
function fJ has
x-intercepts at
at
-‐33 and
and 2, and
12, which
which of
of the
and ay-intercept
a y-intercept at
at 12,
the
following could
following
could define
define f ?
?
The hmction f is defined by J(x) = 3x + 2 and
the function
function g is defined
the
by g(x)
defined by
g(x) =
f(2x) -‐ 1.
= J(2x)
l.
What is the
What
the value
value of
of g( 10)
lO) ?
?
A) fJ(x)
(x + 3)2(x -‐ 2)
A)
( x )‐=‐ (x
(+3)2<x
z)
B)) f ((x)X =) ((xH+
+ 33)(x
B
X ‐ )-( 2)
2)22
C)) fJ(x)
+ 2)
C
( X ) = (x
( X-‐ 3)2(x
3)2 (X+2)
D) ) fJ ((x) =
= ‐(x( X- 3)(x
D
3)(X +
+ 2)
2)22
If J(x)
= 16 +x x
2
J(x) = x 2 + 1
2
is the
the value
value
for all x 7é0,
-:fa0, what
what is
g(x)=x 2 - 1
g(X)=x2‐1
off (- 4)?
The functions
The
defined above.
above. What
What is
is
functions f and
and g are
are defined
the
the value
value off(g(2))
of f(g(2))? ?
A) - 8
8) -4
C) 4
A) 3
D) 8
B) 5
C
0
C)) 110
D
7
D)) 117
X
0
1
2
J(x)
-2
3
18
In the
the xy-plane,
xy-plane, which
which of
of the
the following
following
translati
translations
on s of
of the
the graph
graph of
of y
y = 2x2
2x 2 ‐- 2
2 results
results in
in
2
the
the graph
graph ofy
of y =
= 2x2
2x +
+ 44??
Several values
Several
are given
given in
in the
the
values of the
the function
function fJ are
table above.
above . If
table
(x) = ax2
are
If fJ(x)
ax2 +
+ bb where
where a
a and
and b
bare
constants, what
constants,
value of
(3) ??
what is the
the value
offf (3)
A)
A)
8)
B)
C)
D)
A
3
A)) 223
B)
B) 39
39
C)
C) 43
43
D
6
D)) 556
141
141
A translation
translation 2
2 units
units downward
downward
A translation
translation 6
6 units
units upward
upward
A translation
translation 2 units
units to
to the
the left
left
A translation
translation 6
6 units
units to
to the
the right
right
CHAPTER 16 FUNCTIONS
in the
the xy-plane,
xy-plane, the
the graph
graph of the
the function
function f
[n
reaches its
its maximum
maximum value
value at the
the point
point (3,f (3)
(3) ).
reaches
The function
function g is defined
The
defined by g(x)
g(x) =
= ff(x
(x) +
+ 7.
7. At
which of the
the following
following points
points in the
the xy-plane
xy-plane
which
its maximum
maximum value?
does the
the graph
graph of g reach
does
reach its
value?
y
A
7)
A) (10,f(10)
(10, f (10) + 7)
B)
(f (3), f (3) ++ 7)
B) (f(3),f(3)
C) (3,f(10))
(3,f( lO))
C)
D) (3,](3)
(3,f(3) ++ 7)
D)
The graph
AB
graph of the
the function
function f and
and line
line segment
segment E
are shown
above . For
how many
are
shown in the
the xy-plane
xy-plane above.
For how
many
between -‐33 and
values of x between
values
and 3 does
does J(x)
f(x) =
= c?
c?
2
fJ(x)
(x) = ✓xx ‐ 2
The function
function f is defined
defined above
above for all x 2:
The
2 2.
2.
Which of the
the following
following is
is equal
equal to
to
Which
f(18) -‐ f f(1
f(18)
( 11l ))??
A)
A)
B)
B)
C)
C)
D)
D)
X
f(x)
f<x>
-4
3
5
- 2
0
2
3
4
f (3)
f(3)
ff(5)
(5)
f(6
f(6))
ff(7)
(7)
The function f is defined by J(x ) = (x - 3)2,
and the function g is defined by
g(x) = x2 + 4x + 4. The graph of g in the
xy-plane is the graph off shifted k tmits to the
left. What is the value of k ?
2
16
4
8
The table above gives some values for the
function f. If g(x) = 2f(x), what is the value of
kif g(k) = 8?
A) 2
B) 3
C) 4
D) 8
142
142
THE
THE COLLEGE
COLLEGE PANDA
PANDA
A calculator is allowed on the following
questions.
Ifg(c)
If g(c) := 5,
5, what
what is
is the
the value
value of
of f(c)
J(c)??
A
A)) -‐ 22
B) 3
_ xx +
+1
l
y=
x
‐ x ‐ 11
C) 5
D) 6
Which
Which of the
the following
following points
points in
in the
the xy-plane
xy-plane is
is
NOT
NOT on
on the
the graph
graph of y
y??
A
1
A) (- 2, 3)
8) (- 1, 0)
1
.
C) (0, - 1)
l f J(x)
f ( x ) = -‐ 33xx + 55 and
and éfla)
10, what
what 1s
i sthe
If
J(a ) = 10,
the
D) (1,2 )
va lue of a
a??
value
2
A)) -‐ 8
A
B)) -‐ 5
B
C) 5
Let the
by g(x)
g(x) =
: @.
the function
function g be
be defined
defined by
ffx.
g(a) =
= 6, what
what is th
thee value
value of a ??
If
D)) 8
D
A) 3
B)
8) 6
C) 9
X
D)
D) 12
12
V --------~-------
Questions
following
Questions 20-21
20-21 refer to
to the following
0
1
2
3
information.
information.
4
X
- 2
- 1
0
1
2
3
f(x)
3
5
- 2
3
6
7
J(x)
-4
-8
3
6
g(x)
5
7
2
4
6
4
7
5
2
-3
Several
Several values
values of the
the function
function f are
are given
given in the
the
table
above.
If
the
function
g
is
defined
table above . If the function
defined by
g(x)
value ofg(3)
g(x) =
= f(2x
J(2x ‐- 1),
1), what
what is
is the
the value
of g(3)??
5
7
1
A) 2
A)
The
3 are
are defined
the six values
values
The functions
functions f and
and g
defined for the
of x shown
shown in the
the table
table above.
above .
B)
8) 6
C) 5
D) 7
D)
What is the value of J(g( - 1)) ?
A) - 2
B) 3
C) 5
D) 6
143
CHAPTER 16
FUNCTIONS
CHAPTER
16 FUNCTIONS
IIff f(x)
f ( x ) is aa linear
f(2) :::;
S /(3),
f(3)
linear function
function such
such that
that /(2)
f(4)
f(6) = 10, which
which of
/(4) 2
~ f(5),
/(5), and
and /(6)
of the
the
following
following must
must be
be true?
true?
f(x) :=44xx -‐ 3
J(x)
g(x) = 33x+5
x+5
g(x)
The functions
The
defined above.
Which
functions f and
and g are
are defined
above. Which
the following
of the
is equal
following is
equal to
to f (8)
(8) ?
?
A) ) f/(3)
A
( 3 ) < f(O)
f ® ) <<f/(4)
@)
B)
f(O)
=
0
B) f(0 >=
C) ) fl
C
m >>110
0
f(O)
D)
D ) ff(O)
( 0 )=
‐‐ 10
10
A) g(l)
B) g(3)
C) g(5)
D) g(B)
y
y
--
Y = g(x)
-
1
+-----::Q +----------1----+
X
1
'
.
•--~
I
!
•
····-
I ..
The graph of the function g is shown in the
xy-plane above, and the function f (not shown)
is defined by f (x) = x3 . If g is defined by
g(x)
(x +
a) + b, where
are constants,
g(x) := fJ(x
+a)+
where a and
and b
bare
constants,
what
what is the
the value
value of a
a+
+ bb??
The graph
graph of f(x)
The
f (x) is shown
shown in the
the xy-plane
xy-plane
above.
If g(x)
= (x +
3)(x
‐
1),
for
above. If
g(x) =
+ 3)(x - l ), which
which of the
the
following
( x ) > g(x)
following values
values of x is
is ff(x)
g(x)??
A
A)) ‐- 3
A
A)) ‐- 5
B)) -‐ 2
B
B
B)) -‐ 1
C) 1
C) 1
D) 2
D)) 5
D
In the xy-plane, the graph of the function g is the
image of the graph of the function f after a
translation of 11;~ units
translation
right. Which
units to the
the right.
Which of the
the
following
g(x) ?
following defines
defines g(x)
?
A) ) g(x)
A
g u )==J(3x
f 6 r-‐ 2)
E
B)) gg(x)
f(3x + 2)
B
( x ) = fGX+fl
C)) gg(x)
3)
C
( x )=
= ff(2x
o ‐- w
D)
g(x)
=
f(2x
3)
D ) g u ) = f o ++%
144
THE COLLEGE
PANDA
THE
COLLEGE PANDA
w”
y
y
A
2
The
the function
is shown
The graph
graph of the
function y =
= 9 -‐ xx2
shown in
AB ??
the xy-plane
the length
the
xy-plane above
above.. What
What is the
length of E
The function f(x) = x3 + 1 is graphed in the
xy-plane above. If the function g is defined by
g(x) = x + k, where k is a constant, and
J(x) = g(x) has 3 solutions, which of the
following could be the value of k?
A)
3v'2
A) 3\/§
B ) 3v'10
3m
B)
C) 9
D ) 9v'10
9m
D)
A)) -‐ 1
A
B) 0
C) 1
D) 2
y
[n the
-+- 12,
12, where
where
1n
the xy-plane,
xy-plane, the
the function
function y = ax
ax +
constant, passes
through the
the point
point ((‐a,a).
- a, a) .
a is a constant,
passes through
If a >
value of a ??
> 0, what
what is the
the value
The function f is graphed in the xy-plane above.
If the function g is defined by g(x) = f(x) + 4,
what is the x-intercept of g(x)?
A) - 3
B) - 1
C) 3
D) 4
145
17
Quadratics
Quadratics
Just
as lines
were one
group of
functions that
that have
o w n properties,
quadratics are
are another.
another. A
A quadratic
quadratic
Just as
lines were
one group
of functions
have their
their own
properties, quadratics
is a
function in the
is
a function
the form
form
f(x) =
r - ax
axz2 + bx
b x+
J(x)
+c
in which
which the
highest power
power of x is
is 2.
2. The
The graph
graph of
of aa quadratic
quadratic is
the highest
is a
a parabola.
parabola .
To review
we’ll walk
walk through
through aa few examples
various properties
properties you
you need
need to
to
review quadratics,
quadratics, we'll
examples to
to demonstrate
demonstrate the
the various
know .
know.
QUADRATIC 1:
QUADRATIC
2 -- 44x
f(x) = xx2
x ‐- 21
f(x)
21
The
Roots
The Roots
The
The roots
roots refer
refer to the
the values
values of x that
make Jf ((x)
x) =
: 0. They're
They’re also
x‐intercepts and
and so
solutions.
We’ll
that make
also called
called x-intercepts
lutions . We'll
mainly
use
the
mainly use the term
term "root"
“ r o o t ” in this
this chapter,
the other
other term
termss are
just as
Don’t
forget that
that they
they all
all
chapter, but
but the
are just
as common.
common. Don
't forget
mean the
the same
mean
thing.. Here
Here,, we
we can
just factor
find the
roots:
same thing
can just
factor to
to find
the roots:
2 - 4x
:00
xx2~4x‐21
- 21 =
( . r-‐ 77)(x
) ( . r+
+ 3)
3 ) =2 0
(x
x
X
: 7,
=
7, -~33
The roots
roots are
are 7 and
The
this means
means the
the quadratic
quadratic crosses
the x-axis
x-axis at
at xx =z 77 and
and x.r =: -A3.
and -43.
3. Graphically,
Graphically, this
crosses the
3.
The Sum
Sum and Product of the Roots
Roots
=
We
already found
We already
found the
so their
u m is just
just 77 + ( -‐ 33)) z 4
their product
product is
is just
just 77 xx -‐33 =
z -‐ 221.
1 . This
This
the roots,
roots, so
their ssum
4 and
and their
was really
easy, so
was
really easy,
about these
these values?
values? Because
Because sometimes
sometimes you'll
you’ll have
have to find the
the sum
s u m or
or the
the
so why
why do we care
care about
product
the roots
roots without
without knowing
knowing the
the roots
roots themselves.
themselves. H
o
w
do
we
do
that?
product of the
How do we do that?
.
.
.
b
Given
quadratic of the
form y =
: ax
axz2 +
+ bx + c, the
the sum
s u m of
of the
of the
the roots
roots
Given a
a quadratic
the form
the roots
roots IS
is equal
equal to
to -‐ ~E and
and the
the product
product of
a
,
c
. equal
c.
is
1s
equa 1to -E.
a
146
THE COLLEGE
COLLEGE PANDA
THE
PANDA
our example,
example, a =
In our
: l,Lbb =
: -‐ 44,, c = -‐21.So,
21. So,
Sumz‐é=‐‐:4
= - -- 4 = 4
a
1
l
b
Sum = - -
C
Product
Product = -E = --‐ ‐-_21fl = ‐- 2211
1
l
a
See how
were able
these values
values without
without knowing
knowing the
roots that
that we
we
how we were
able to determine
determine these
the roots
roots themselves?
themselves? The
The roots
found
found earlier
earlier just
just confirm
confirm our
our values.
values.
The Vertex
The
The vertex
The
vertex is the
the midpoint
midpoint of a
a parabola.
parabola .
y
vertex
vertex
The x‐coordinate
The
vertex is always
always the
midpoint of the ttwo
w o roots,
roots, which
found by
averaging them.
x-coordinate of the
the vertex
the midpoint
which can
can be
be found
by averaging
them.
Because
roots are
are 7
7and
the vertex
vertex is at x =
z
Because the
the roots
and -‐ 33,, the
7+
+ (-( ‐ 33))
z 2. When
When x =
: 2,f(x)
(2)22 -‐ 4(2)
5.
=
2, J (x) := (2)
4(2) ‐- 21
21 := ‐- 225.
2
Therefore, the
Therefore,
vertex is at (2, -‐25).
Note that
that the
maximum or m
i n i m u m of a quadratic
always at
at the
vertex.
the vertex
25). Note
the maximum
minimum
quadratic is
is always
the vertex.
ln this
this case,
in
minimum of ‐25.
case, it’s
it's a minimum
- 25.
Form
Vertex Form
Just asslope-intercept
as slope-intercept form
Just
b)is
of representing
form is
is one
one way
way of
of representing
representing
form (y = mx+
mx + b)
is one
one way
way of
representing a
a line,
line, vertex
vertex form
a quadratic
quadratic function
a
function.. We've
We’ve already
t w o different
different ways
ways quadratics
can be
be represented,
namely standard
already seen
seen two
quadratics can
represented, namely
standard
form (1;
(y =
form
: ax2
form (y =
: (x -‐ a
a)(x
looks likey
= a(x
a(x -‐ h)2
+ k.
ax2 +
+ bx
bx++ c) and
and factored
factored form
)(x -‐ b)). Vertex form
form looks
like y =
h)2 +
To get
get a
a quadratic
quadratic function
function into
into vertex
vertex form
form,, we
we have
have to
to do
do something
something called
called completing
completing the
the square.
square. Let’s
Let's walk
walk
through
through it step-by‐step:
step -by-step:
yy z= x2
x2 -‐ 4x
4x -‐ 21
21
the middle
See the
The -‐ 44.. That
That’s
the key.
key. The
The first step
is to divide
divide it
by 2
Then write
the
middle term?
term? The
's the
step is
it by
2 to
to get
get -‐ 22.. Then
write the
following:
following :
y = (x ‐- 2)2
2) 2 ‐- 21
where we put
See where
? The
The first part
o w the
take that
that -‐22 and
square it.
We get
get
put the
the ‐- 22?
part is done.
done . N
Now
the second
second step
step is to take
and square
it. We
4.
yy:= (x‐2)2‐21‐4
(x - 2)2 - 21 - 4
See where
where we
Wesubtracted
at the
end. The
The vertex
we put
put the
the 4?
4? We
subtracted it at
the end.
vertex form
form is
is then
then
y z= (x -‐ 2)2 -‐ 25
To recap,
recap, divide
middle coefficient
to get
get the
the number
divide the middle
coefficient by
by 22 to
number inside
inside the
the parentheses.
parentheses. Subtract
Subtract the
the square
square of
of
that
at the
the end.
that number
number at
end .
147
CHAPTER 17
QUADRATICS
CHAPTER
17 QUADRATICS
Completing the
the square
square takes
Completing
takes some
and practice,
practice, so
so if
if you
some time
time and
you didn't
didn 't catch
catch all of
of this,
this, first
first prove
prove to
to yourself
yourself
that it is indeed
that
the same
same quadratic
the result.
result. Then
indeed the
quadratic by
by expanding
expanding the
Then repeat
repeat the
the process
process of
of completing
completing the
the square
square
yourself. If you’ve
yourself.
slightly different
different way,
way, feel free to
many m
o r e examples
in this
this
you've been
been taught
taught a
a slightly
to use
use it. We’ll
We'll do
do many
more
examples in
chapter.
chapter.
N
o w why
vertex form?
look at
at the
the numbers!
It’s called
vertex form
reason. The
The
Now
why do
do we
we care
care about
about vertex
form? Well, look
numbers! It's
called vertex
form for
for aa reason.
vertex (2, -‐25)
vertex
be found
found just
just by looking
at the
the numbers
numbers in
found the
the vertex,
25) can
can be
looking at
in the
the equation.
equation. But we
we already
already found
vertex,
you say! Yes, that’s
true, but
had to find the
the roots
roots to
to do
do so
finding the
o t always
you
that 's true,
but we had
so earlier,
earlier, and
and finding
the roots
roots is
is nnot
always so
so
easy. Vertex form
vertex without
knowing the
quadratic.
It’s
also
very
much
form allows
allows us
us to
to find the
the vertex
without knowing
the roots
roots of
of a
a quadratic. It's also very much
tested on the
tested
the SAT!
SAT!
One
o t e ‐ o n e of
o f the
the most
common mistakes
students make
= ((xx -‐ 2)
2)22 -‐ 2
and think
One final n
note-one
most common
mistakes students
make iiss ttoo look
look a
att y =
255 and
think the
the
vertex
is at
(‐ 2,
2, -25)
‐ 2 5 ) instead
instead of
of (2,
(2 -‐ 225).
5 ) One
One pattern
pattern of
of thinking
avoid this
What
vertex is
at (thinking II use
use to
to avoid
this mistake
mistake is
is to
to ask,
ask, What
value of x would
would make
expression inside
parentheses eq11al
equal to zero?
0.
value
make the
the expression
inside the parentheses
zero? Well, x:
x =2
2 would
would make
make xx ‐- 2
2 equal
equal to
to 0.
Therefore,
vertex is at xx‐ =
‐ 2. T
h i sis
thinking you
get the
the solutions
solutions from
from the
Therefore,the
the vertex
This
is the
the same
same type
type of
of thinking
you would
would use
use to
to get
the
factored
factored form
form y‐‐
y = (x ‐- a)(x
a)(x -‐ b).
b).
The Discriminant
The
Discriminant
If
then the
As we
l l explain
explain later,
the
If a
a quadratic
quadratic is in the
the form
form ax2
ax2 + bx + c,
c, then
the discriminant
discriminant is
is equal
equal to
to b2‐
b2 - 4ac.
4ac. As
we '’ ll
later, the
discriminant'
Before we
we explain
its significance,
significance, let’
the
discriminant IisSa
a component
component of
of the
the quadratic
quadratic formula
formula. Before
explain its
let'ss calculate
calculate the
discriminant for oour
discriminant
u r first example,
example,
2
f(x)
f(x) = xx2‐4x‐21
- 4x - 21
Discriminant
Discriminant =
= b2‐
4ac =
: ((‐4)2
: 100
b2 - 4ac
- 4 )2 -‐ 4(1)(‐21)
4(1) ( - 21) =
Now, what
does the
the discriminant
di scriminant mean?
Now,
what does
value of the
doess not
n o t matter.
mean? Well, the
the value
the discriminant
discriminant doe
matter. What
What matters
matters is
is
sign of the
the discriminant‐whether
discriminant -w hether it's
the sign
it’s positive,
positive, negative,
negative, or zero.
care that
zero. In
In other
other words,
words, we
we don’t
don't care
that it’s
it's
100, we
we just
just care
care that
that it's
it’s positive
positive.. Letting
be short
short for discriminant,
discriminant,
Letting D be
W h e nDD =
z O0,,
When
y
W
h e nDD > 00,,
When
y
W
h e nDD < O
When
0,,
y
-----0-+----there are
are ttwo
there
w o real
t w o solutions).
solutions)
real roots
roots ((two
tthere
h e r eis one
one real
real root.
root.
X
there
no real
real roots.
roots.
there are
are no
The
The Quadratic
Quadratic Formula
Formula
we've seen,
As we’ve
seen, the
important aspect
aspect of a
the roots
roots are
are the
the most
most important
a quadratic.
quadratic . Once
Once you
you have
have the
the roots,
roots, things
things like
like
ve rtex form
form and
vertex
and the
discriminant are
are n
o t as
ashelpful.
or work
the discriminant
not
helpful. Unfortunately,
Unfortunately, the roots
roots aren’t
aren't always
always easy
easy to
to find
find or
work
with. That’s
That's when
when vertex
with.
u m / product of the
can get
get us
to the
the answer
vertex form,
form, the
the discriminant,
discriminant, and
and the
the ssum/product
the roots
roots can
us to
answer
faster.
But if
if we
we m
u s t find the
there is always
always one
one surefire
surefire way
way to
o ‐ t h e quadratic
must
the roots,
roots , there
to do
do sso-the
quadratic formula.
formula.
_ ‐b:|:\/b2‐4ac
- b± Jb 2 - 4ac
x=
_T
2a
forax2+bx+c:0.
for ax2 + bx + c = 0.
148
THE COLLEGE
COLLEGE PANDA
THE
PANDA
For the purpose
quadratic formula
purpose of learning,
learning, let’s
let's apply
apply the
the quadratic
formula to
to our
our example,
example,
f(x) = x 22‐- 44xx ‐- 2211
J(x)
According to the
According
the formula,
are
formula, the
the roots
roots //ssolutions
olutions are
_4
m _= 44 x± 1100 _= ?or _
x_
= -‐(‐4)i,/(‐4)?‐4(1)(‐21)
(- 4) ± J( - 4)2 - 4(1)( - 21) =
4 ± i/100
x
‐ T ‐ 2(1)T ‐ ‐ ‐ 27 0 2 r ‐ 33
These
values that
that we
factoring..
These are
are the
the same
same values
we got
got through
through factoring
Notice
4116, is
is tucked
tucked under
under the
o w does
does this
Notice that
that the
the discriminant,
discriminant, 172
b2 -~ 4ac,
the square
square root
root in
in the
the quadratic
quadratic formula.
formula. H
How
this
help us
know about
the discriminant?
help
us understand
understand what
what we
we know
about the
discriminant?
takes effect
effect and
and we
we end
w o different
Well, when
when b2‐
b2 - 4ac > 0,
0, the
the"“ :±l :"" takes
end up
up with
with ttwo
different roots.
roots. When
When 172
b2 ‐- 4ac
4ac =
= 0,
0, the
the
”" ±i "” does
does not
n o t have
we’re essentially
essentially adding
of which
which give
give us
us the
same
have an
an effect
effect since
since we're
adding and
and subtracting
subtracting 0,
0, both
both of
the same
root.. When
square root
root of
gives us
us
root
When [72
b2 ‐- 4116
4ac < O,
0, we’re
we're taking
taking the
the square
of a
a negative
negative number,
number, which
which is
is undefined
undefined and
and gives
no real
no
real roots
(we’ll talk
number in
in aa later
roots (we'll
talk about
about imaginary
imaginary number
later chapter).
chapter).
Hopefully,
you understand
understand where
meanings come
come
Hopefully, the
the quadratic
quadratic formula
formula helps
helps you
where the
the discriminant
discriminant and
and its
its various
various meanings
from. Understanding
from.
will help
you remember
remember the
Understanding this
this connection
connection will
help you
the concepts.
concepts.
Now that
Now
we’ve taken
you on
tour through
that we've
taken you
on aa thorough
thorough tour
through the
the properties
properties of
of quadratics,
quadratics, we’ll
we'll go
go through
through a
a few
few
more examples
more
pace.
examples to illustrate
illustrate some
some important
important variations,
variations, but
but we’ll
we'll do
do so
so at
at a
a much
much faster
faster pace.
QUADRATIC 2:
QUADRATIC
2:
ff(x)
(x) =
: -x
‐x22 ++ 66xx‐- 1100
The Roots
Roots
This quadratic
n d in fact,
fact, if
if we
we look
look at
the discriminant,
quadratic cannot
cannot be
be factored.
factored. A
And
at the
discriminant,
ac =
: (6)
(6)22 -- 44(-1)(
( ‐ 1 ) ( ‐-110)
0 ) := ‐-44
b1922 -‐ 44ac
it's negative,
negative, which
it’s
are no
no real
real roots
roots or solutions.
solutions. The
The graph
the quadratic
which means
means there
there are
graph of
of the
quadratic makes
makes this
this even
even
more
more clear:
y
2 term
When the coefficient
coefficient of the
When
x2
negative, the
the parabola
shape of
U.”
the x
term is negative,
parabola is
is in
in the
the shape
of an
an upside-down
upside-down ”"U."
149
CHAPTER 17
CHAPTER
17 QUADRATICS
QUADRATICS
The
The Sum
Sum and
and Product
Product of the Roots
Roots
x ) = -‐x2
10
Jf ((x
x 2 +66xx -‐ 10
b
6
Sum = - - = - - = 6
a
-‐11
a
C
- 10
Product
P r o d u c=t : -5 =
z -_‐10 :=110
0
a
-‐11
Sum= ‐9 = ‐i =6
what!? We already
already determined
determined that
that there
there were
were no
no roots
roots.. How
H o w can
sum and
and aa product
Wait, what!?
can there
there be
be a
a sum
product of roots
roots
that
don’t exist?
exist? Well, the
the quadratic
quadratic doesn't
doesn’t have
have any
any real roots,
have ima
imaginary
roots. The
values
roots, but
but it does
does have
ginary roots.
The values
that don't
above are
are the
the sum
sum and
and product
product of these
these imaginary
imaginary roots
roots.. We'll
We’ll cover
numbers in a
above
cover imaginary
imaginary numbers
a later
later chapter.
chapter.
Vertex Form
Because the roots
roots are
are imaginary,
imaginary, we
we can't
can't use
use their
their midpoint
midpoint to find
these cases,
cases, we
we must
must get
get the
the
Because
find the
the vertex.
vertex . In these
quadratic
quadratic in vertex
vertex form.
form. We'll
We’ll have
have to complete
complete the
the square
square..
yyz‐x2+6x‐10
= - x 2 + 6x - 10
First, multiply
multiply everything
everything by negative
negative 11 to get
get the
the negative
negative out
out of the
Having the
the negative
negative there
makes
First,
the x2
x 2 term.
term . Having
there makes
later .
things
needlessly complicated
ll multiply
back by -‐11 later.
things needlessly
complicated.. We'
We’ll
multiply everything
everything back
-‐y:x2‐6x+10
y = x 2 - 6x + 10
get 9. Remember
put the
the -‐33 inside
the
Divide
term by 2 to get -‐33 and
Divide the
the middle
middle term
and square
square this
this result
result to get
Remember that
that we put
inside the
parentheses
pieces in place,
place,
parentheses with
with x and
and subtract
subtract the 99 at
at the
the end.
end. Putting
Putting these
these pieces
= (x‐3)2+10‐9
-‐ y =(
x - 3) 2 + 10 - 9
= (xx -‐ 3)
3)22 +
+11
-‐yy =(
N o w multiply
everything by
by -‐11 again,
again,
Now
multiply everything
: -‐ ((xx -‐ 3)
3)22 -‐ 1
y=
Now
the vertex
the graph
graph is an
upside-down ”U,”
"U," -‐11 is the
the
Now it's
it’s easy
easy to see that
that the
vertex is at
at (3,
(3, -‐ 11)).. And
And because
because the
an upside-down
maximum
value of Jf(x).
(x).
maximum value
QUADRATIC
QUADRATIC 3:
3:
2
f ( x ) = 2x
2x2+5x‐3
f(x)
+ 5x - 3
The Roots
Roots
The
Wecan
can factor this
quadratic to get
get
We
this quadratic
2x22 +
+ 5x -‐ 3 =
= 0
2x
((2x
2x -‐ 1l ) ( Xx + 33)) = O
0
x
X
z 0.5, -‐33
=
The roots
roots are
are 0.5 and
you don't
don’t know
know how
how we
we factored
factored this,
teaching factoring
the
The
and -‐ 33.. If you
this, unfortunately
unfortunately teaching
factoring from
from the
ground
up is not
n o t within
the scope
scope of this
this book.
book. Don't
Don’t be
be afraid
afraid to look
look up
factoring lessons
lessons and
and drills
ground up
within the
up factoring
drills online
online
and in your
your textbooks.
textbooks. It's
It’s an
an essential
essential skill
skill to have
have.. Just
Just know
o u t there
there involves
involves a
and
know that
that every
every method
method out
a little
little
trial
and
error.
And
if
you’re
ever
stuck,
the
quadratic
formula
is
always
an
option.
trial and error. And you're ever stuck , the quadratic formula always an option .
150
THE COLLEGE
COLLEGE PANDA
PANDA
THE
The
The Sum
Sum and
and Product
Product of the
the Roots
Roots
5
b
Sum = -‐B = -‐§ =
: -‐2.5
Sum
2.5
a
a
2
2
C
-3
Product = E = _‐3 = -‐ l1.5
.5
Product
22
a
a=
The
The Vertex
Averaging the
the two
t w o roots
roots to find
find the
x-coordinate of the
the vertex,
Averaging
the x-coordinate
vertex,
0 5+
+ (( ‐ 33)) =_ -‐2.5
0.5
2.5 =_ _ 25
_ 2
22
2 _ 1.25
1.
Plugging
Plugging this
this into
into f(x)
f (x) to find
find they-coordinate,
the y-coordinate,
2(‐1.25)2
:
ff(‐1.25)
(- 1.25) = 2(
- 1.25)2 + 5(‐1.25)
5( - 1.25) -‐ 3
3 =
-‐6.125
6.125
The vertex
vertex is at
at (( ‐ l 1.25,
. 2 5 , -‐6.125).
Because the
the quadratic
quadratic opens
shape of aa "U,"
” U , ” the
the minimum
The
6.125). Because
opens upward
upward in the
the shape
minimum
value
is -‐6.l25.
6.125.
value of J(x)
f ( x ) is
Vertex Form
: 2x
2x22 +
+ 5x -‐ 3
y=
2
First,
square, always
make sure
sure the
the coefficient
coefficient of xx2
First, divide
divide everything
everything by 2. Before
Before completing
completing the
the square,
always make
is l. We'll
We'll
multiply
multiply the
the 22 back
back later
later..
y
5
3
2
2=X
+ 2X- 2
25
Divide
result to get
get~!.
We put
put the
the~2 inside
the parentheses
with
Divide the
the middle
middle term
term by 2 to get~
get 2 and
and square
square this
this result
1‐6. We
inside the
parentheses with
25
16
x and
and subtract
subtract the
the 3‐2 at
at the
the end.
end.
-"‐<+?)2
E‘
x 4
2 16
z_
!!..
= (x+ §Y_9
~)2
- 4916
22‘(x+4
4
16
Multiplying by 2,
Multiplying
I/
(x+ ~)
2 - 49
‐2(x+§)2‐9
4
8
=2
·3“
+125)2
yi, = 2(x +
1.25) 2 -‐ 6.125
This
consistent with
This is consistent
with the
the vertex
vertex found
found above
above..
The
Discriminant
The Discriminant
For
let's calculate
will confirm
the fact that
this
For the
the sake
sake of completeness,
completeness, let's
calculate the
the discriminant.
discriminant. Hopefully,
Hopefully, it will
confirm the
that this
quadratic
has two
t w o distinct
distinct real
real roots
roots..
quadratic has
_
2x22+
‐ 2x
+ 5x
5x -_ 3
y=
3
2 - 4ac := (5)
Discriminant =
: b62‐
(5)22 -~ 4(2)(‐3)
Discriminant
4(2)(-3) = 49
The
that this
this quadratic
quadratic has
has ttwo
roots.
The discriminant
discriminant is positive,
positive, which
which confirms
confirms the
the fact that
w o real
real roots.
151
151
CHAPTER
CHAPTER 17 QUADRATICS
QUADRATTCS
QUADRATIC
QUADRATIC 4:
f(x)
4x2 - 12x + 9
f(x) ==4x2-12x+9
The
The Roots
Roots
We
ld factor
let's use
the quadratic
Wecou
could
factor this, but
but let's
use the
quadratic formula
formula instead.
instead.
b ± Jb 2 - 4ac _ -‐(‐12):t\/(‐12)2‐4(4)(9)
x _ -‐bi\/b2‐4ac
1 2±1 \/6
(- 12) ± J( - 12)2 - 4(4)(9 ) _ 12
J6 _ 3-3
x =-----_
2a
2a
'
2(4)
_
8
‘:'z2
As
one root,
root,~As you
you can
can see,
see, the discriminant
discriminant is 0 and
and the quadratic
quadratic has
has just one
2
2
The Sum
Sum and Product
Product of the Roots
Roots
- 12
b
Sum = -‐~- = -‐‐ z 3
Sum
=
a
4
C
9
a
4
Product
Product = - = -
If we
we only
have aa sum
and a
a product
are they
they different
different
only have
have one
one root,
root, how
how is it that
that we
we can
can have
sum and
product of two
t w o roots?
roots? Why
Why are
from
found?
from the
the one root
root we found?
Here's
a quadratic
one root,
really has
has ttwo
that are
are the same.
same.
Here’s the thing.
thing. While we
we may
may say a
quadratic has just
just one
root, it really
w o roots
roots that
2 term, is expected to have two roots. When they're the same, we just refer to
After
After all,
all, aa quadratic,
quadratic, with an
an xx2
term, expected
have t w o roots. When they’re the same, we just refer to
them
them as
as one
one..
9
So
add them,
and if we
we multiply
multiply them,
them , we
Soour
o u r "two"
” t w o " roots
roots are
are ~3 and
and ~.
g. If we
we add
them, we
we do
do indeed
indeed get 3,
3, and
we do
do get ~.
Z.
The Vertex
The
When aa quadratic
root, the
vertex is the same
as the
because aa
When
quadratic has
has just
just one
one root,
the x-coordinate
x‐coordinate of the
the vertex
same as
the root.
root. That's
That’s because
quadratic
quadratic is tangent
tangent to the x-axis when
when it has
has one
one root.
root.
y
0
Q
2
3
They-coordinate
course, 0. Therefore,
( ~, O).
0). The minimum
minimum value
The y-coordinate is, of course,
Therefore, the
the vertex
vertex is at
at (5,
value off
of f (x)
(x) is 0.
0.
152
THE
COLLEGE PANDA
THE COLLEGE
PANDA
Vertex Form
= 4x
4x22 -‐ 12x +
+9
y=
2
First, divide
square, always
always make
make sure
the coefficient
coefficient of xx2
is 1. We'll
First,
divide everything
everything by 4.
4. Before
Before completing
completing the
the square,
sure the
We’ll
multiply the
back later.
multiply
the 44 back
later.
y1 : 22 _
9
-4 = Xx - 3x+4
3X + -2
4
. .
.
4
3
9
2
4
Divide the
result to get
get ~.
We put
the -‐%~ inside
the parentheses
parentheses
D1v1de
the middle
middle term
term by
by 22 to get
get -‐ ~5 and
and square
square this
this result
1. We
put the
inside the
2
with
with x and
and subtract
subtract the
the ~g at
at the
the end.
end.
The
The constants
constants cancel
cancel out.
out.
Multiplying by
by 4,
4,
Multiplying
This
consistent with
found above.
This is consistent
with the
the vertex
vertex found
above.
Wow! We
much everything
to know
about quadratics
Wejust
just covered
covered pretty
pretty much
everything you
you need
need to
know about
quadratics.. Unfortunately,
Unfortunately, we're
we’re not
not
quite done
as there
you should
should be
exposed to.
quite
done yet
yet as
there are
are aa few tough
tough question
question variations
variations that
that you
be exposed
EXAMPLE
the xy-plane,
y = x2
x 2 ‐- Sx
EXAMPLE 1: In the
xy-plane, the
the parabola
parabola with
w i t h equation
equation 31=
5x +
+ 6 intersects
intersects the
the line
line y =
= 3x -‐ 10
b). What
the value
at
at point
point (a,
(a,b).
What is
is the
value of b?
b?
This
the systems
systems of equations
equations chapter,
chapter, but
reviewing it
This is aa question
question type
type that
that we
we already
already covered
covered in the
but we're
we’re reviewing
again here
next few examples.
examples. The
concept is
whenever
again
here because
because it will
will help
help you
you understand
understand the
the next
The core
core concept
is that
that whenever
you
have to find
intersection point(s)
solve the
the system
system consisting
their equations.
equations. The
you have
find the
the intersection
point(s) of two
t w o graphs,
graphs, solve
consisting of their
The
solutions
we have
have
solutions to the
the system
system are
are the
the intersection
intersection points.
points. Here,
Here, we
yy = x2
x 2 -‐ 5x
Sx +
+66
y = 3x
3x -‐ 10
10
Substituting
into the
Substituting the
the second
second equation
equation into
the first,
first,
3x‐10zx2‐5x+6
3x
- 10 = x 2 - Sx + 6
00 == x¥2 -‐ &
Bx+
+m
16
: ((x
x ‐- 44)F2
00 =
x=4
x=4
To find
find they,
equations: y = 3(4)
the y, we
we plug
plug x =z 44 into
into either
either of the
the original
original equations:
3(4) -‐ 10
10 =z 2.
2. Therefore,
Therefore, the
the point
point of
intersection is at
at (4,
(4, 2) and
and b =
= [I].
I.
intersection
153
CHAPTER
QUADRATICS
17 QUADRAncs
CHAPTER 17
EXAMPLE 2: How man
'
the gra h of =
_ _ 2
.
‑
EXAMPLE2: How manyy times does the
the
10 in the
y = 10in
line 31=
the line
intersect the
3 mm
6x + 3
y _ - xx2 + 6x
p Ofy
a calculator.)
cannot use acalculator.)
xy-plane?(You cannotuse
xy-plane?
graph
We’re
dealing with the intersection of two graphs again. Sowhat do we do?">
We:re deal~g with the intersection of two gra hs a ain. So wh
of
consisting Of
the system
solve the
We solwg
we do . We
at do
th fj
ti~ . g the
eguat10ns.
their
the" equations, Substih1ting
Substituting the
the second
second eq
equation
system C°"5i5ti"8
get
we get
rst, we
into e first
ua on into
1 0=: -‐x2+6.r+3
x + 6x + 3
10
2
+ 6x
6x‐- 7
7
x2 +
= -‐x2
00z
Now
and finish
this to
intersection point(s)
just like
previous
the previous
in the
did in
we did
like we
point(s) just
the intersection
find the
to find
solving this
finish solving
~,ead and
go ahead
couldgo
we could
Now we
example,
but
there’s
a
faster
way.
For
the
purposes
of
this
question,
we
don
’t
care
where
the
intersection
points
points
intersection
where
care
't
don
we
question,
this
of
purposes
the
example, but theres a faster way. For
are.
We
just
want
to
know
how
many
there
are.
there
many
how
know
want
just
We
are.
that.
the discriminant
Sound
discriminant to do
do that.
use the
can use
We can
familiar? We
Sound familiar?
2
4(: (6)
(6)22 ‐- 4
( ‐ 11)() ( ‐ 77)) := 8
Discriminant:
b2‐
- 4ac =
Discriminant = b
there are 2
above . If there
up above.
we set up
equation we
are 2 solutions
there are
The discriminant
which means
solutions to the equation
means there
positive, which
discriminant is positive,
bother finding
didn't bother
points . To summarize,
there must
solutions
must be
be [I} intersection
intersection points.
summarize, we didn’t
finding
above, there
equation above,
the equation
solutions to the
might've
points
intersection
the
and
care,
we
all
for
100
=
x
and
2
=
been x
could've been
the two
and
and the intersection points might’ve
values of x. They could’ve
two values
the
used the
we used
and we
them, and
of them,
were ttwo
there were
that there
mattered was
doesn't matter.
). It doesn’t
been
and (100,6
(100,6).
matter. What
What mattered
was that
w o of
(2,5 ) and
been (2,5)
And
point.
intersection
one
be
only
would
that. If the discriminant
discriminant
determine that.
discriminant were
were 0, there
there would
be one intersection point. And ifif
discriminant to determine
.
points
intersection points.
no intersection
be no
would be
there would
than 0, there
were less than
discriminant were
the discriminant
points
intersection points
the intersection
where the
out where
and figure out
back and
understand this question
you understand
Make sure
Make
sure you
question.. Feel free to go
go back
so
was so
discriminant was
the discriminant
why the
That 's why
formula . That’s
need the
fun . You'll
not fun.
It's not
(Hint: It's
are (Hint:
actually are
actually
You’ll need
the quadratic
quadratic formula.
helpful).
helpful).
EXAMPLES:
EXAMPLE3:
y -‐ kk =
= 0O
= x2-3x + 1
yy=x2‐3x+l
system
the system
does the
k does
of It
the following
of the
For which
constant. For
a constant.
is a
k is
above, k
equations above,
system of equations
the system
Jn the
in
which of
following values
values of
solutions?
equations have
of equations
have no
no real
real solutions?
A) ‐22
A)-
B) -‐11
C)O
D) 1
equation ,
second equation,
the second
into the
equation and
we get y =
First, we
First,
z k from the first equation
and substitute
substitute this
this into
kk = x2
+1l
3x +
x 2 -‐ 3x
O=z x22-‐ 33xx + ((11-‐ kk))
O
The
real solution
no real
have no
should have
above should
equation above
the equation
then the
solution, then
real solution,
no real
has no
equations has
system of equations
If
If the system
solution.. The
than 0.
be less than
discriminant should
discriminant
should be
4k
4k =
4(1)(1 -‐ k) = 9 ‐- 4 +
3)2 -‐4(1)(1
b2 -‐ 44ac
Discriminant = 172
Discriminant:
a c = ((‐3)2
+4k
z 55 +
+4k
answer
2, answer
Only -~2,
negative. Only
being negative.
4k being
+ 4k
5+
in 5
which one
to see
choices to
each of the
test each
Now
N
o w we test
the answer
answer choices
see which
one results
results in
discriminant.
negative discriminant.
a negative
produces a
~ produces
back
Go back
SAT. Go
the SAT.
on the
see on
might see
you might
questions you
toughest questions
the toughest
far showcase
so far
done so
've done
examples we
The examples
we've
showcase some
some of
of the
them .
understand them.
make sure
and make
and
sure you
you understand
154
THE
COLLEGE PANDA
THE COLLEGE
PANDA
EXAMPLE
n) = ‐100n2
- 10Qn2 +
000n to model
model the population
EXAMPLE 4: A biologist
biologist uses
uses the
the function
function p(
p(n)
+ 1,
1,000n
population of seagµlls
seagulls
equivalent forms of p(n)
p(n) displays
on aa beach
beach in year number
number n, where
where 1 ~
5 n ::;
S 10. Which of the
the following equivalent
displays
which the population
reaches that
the
the maximum
maximum population
population of seagulls
seagulls and
and the
the number
number of the year in which
population reaches
that
maximum
maximum as
as constants
constants or coefficients?
‘
A)
4n(25n -‐ 250)
A) p(n) = -‐4n(25n
B)
10(10n2 -‐ 100n)
B) p(n)
p(n) = -‐10(10n2
10011)
C) p(n) = -‐100(n
lOO(n-‐ 5)2 + 2,500
C)
D)
lOO(n- 7)
4,900
D) p(n) = -‐100(n
7)22 +
+4,900
Anytime
maximum or minimum
output (i.e.
(i.e.
Anytime you
you see
see a quadratics
quadratics question
question that deals
deals with
with the maximum
minimum of a function
function output
the y-value),
either
figure
out
the
vertex
or
look
for
vertex
form.
After
all, the
the maximum
y-value), either figure
look
After all,
the vertex
vertex is where
where the
maximum
or minimum
minimum occurs
occurs.. In fact,
fact, the
the answer
answer is
is either
either (C)
(C) or (D) because
those are
are the only
only ones
vertex form
form..
because those
ones in vertex
Furthermore, with
that (D) does
does not
the original
original equation
Furthermore,
with a little
little calculation,
calculation, it's
it's easy
easy to see that
n o t expand
expand to be
be the
equation,, so
so
the
the answer
answer is (C).
However,
learning purpo
ses (and
questions), I'll
this question
question in ttwo
However, for learning
purposes
(and for the
the tougher
tougher questions),
I’ll show
show you
you how
how to do
do this
wo
different ways
ways.. We
We can
can find the
the vertex
vertex using
using the
the average
average of the roots
roots and
vertex form
form..
different
and then
then reverse
reverse engineer
engineer vertex
Or we
transform the
the equation
we can
can transform
equation into
into vertex
vertex form
form directly.
directly.
Solution
and factor,
Solution 1: To find the roots,
roots, we
we set
set the
the equation
equation equal
equal to O
0 and
0
-400,12
100n 2 + 1,000n
1,000n = 0
-‐100n(n
lO0n(n -‐ 10) =
z 0
: 0,10
n=
0, 10
The roots
vertex is 5. Now
roots are
are O
0 and
and 10, which
which means
means the
the x-coordinate
x-coordinate of the vertex
Now we
we can
can plug
plug 5 into
into p(n) to find
find
the
y-coordinate..
they-coordinate
: -‐100(5)2
+1,000(5)
: 2,500
2,500
p(5) =
100(5) 2 +
1, 000(5) =
So
2500). Now
So the
the vertex
vertex is at (5,
(5,2500).
N o w remember
remember what
what vertex
vertex form
form looks
looks like: y =
: a(x -‐ h)
h)22 +
+ k. Given
Given our
o u r values,
values,
we have
have
= a(n -‐ 5)
5)22 +
+ 2,500
p(n) =
2,500
We now
another point
point to work
work with
easy to see that
n o w need
need to find what
what a is. To do that,
that, we
we need
need another
with.. Well, it's
it’s easy
that p(n)
passes
). Plugging
passes through
through the
the point
point (0,0
(0, O).
Plugging that
that in,
in,
0=
z a(O
a(0 -‐ 5)2 +
+ 2,500
2,500
o
2 25a
2511 +
+ 2,500
0=
2,500
2,500
-‐25a
25a = 2,500
a == -‐100
100
[@[J
.
Finally, p(n) = -‐100(n
5)22 + 2,
2,500.
Answer (C) .
Finally,
lO0(n - 5)
500. Answer
155
QUADRATICS
17 QUADRATICS
CHAPTER
CHAPTER 17
divide
First, divide
directly . First,
form directly.
vertex form
the square
involves completing
method involves
second method
This second
Solution
completing the
square to get the vertex
Solution 2: This
l.
coefficient of 1n122 is 1.
the coefficient
ensure the
- 100 to ensure
everything by ‐100
everything
p(n)) =
1,00011
+ 1,00011
- 100n2 +
= ‐100n2
p(n
POI)
_
2 - lOn
p(n) =
‐‐_100
‐n n2_
lOn
- 100
lOn.
be -‐10n.
would be
term would
"middle" term
end, the ”middle"
what to do next?
you remember
Do you
remember what
next? If
If we wrote
wrote the
the constant
constant 0 at the end,
and the
with n and
parenthe ses with
the parentheses
inside the
belong s inside
- 5 belongs
that to get
and square
get -‐55 and
the ‐- 110
Divide the
Divide
0 by 2 to get
square that
get 25. The
The ‐5
end .
the end.
at the
subtracted at
25 gets subtracted
25
p01)
_ (n
25
(n -‐ 5)2
p(n) =
W
_
5)2 ‐- 25
- 100
- 100.
back by ‐100.
everything back
multiply everything
can multiply
we can
Now
N
o w we
2,500
+ 2,500
p(n) = ‐100(n
5)22 +
- l00 (n -‐ 5)
.
[@IJ
that the
again, we prove
And again,
And
prove that
the answer
answer is (C) .
156
THE COLLEGE
PANDA
COLLEGE PANDA
THE
Review:
Review:
+ c,c,
ax2 + bx
form, y =
the form,
Given
= ax2
bx+
quadratic of the
Given aaquadratic
ways:
following ways:
the following
be found
The
roots, also
also called
called solutions
solutions and
and x-intercepts,
x‐intercepts, can
can be
found in the
The roots,
0
Factoring
• Factoring
0
on the
the calculator
calculator (look
for the
the x-intercepts)
x‐intercepts)
(look for
Graph on
• Graph
o
The quadratic
formula x =
quadratic formula
• The
m
-‐ bb± ✓bi2 - 4ac
2a
211
= - -ab
Sum of the
the Roots
Roots = ‐5
Sum
C
Product of the
Roots =
= -2
the Roots
Product
a
2 - 4ac
ThediscriminantD
=b
b2‐4ac
The discriminant D =
solutions.
0
When D >
> 0,
w o real
real solutions.
are ttwo
there are
0, there
• When
one real
When D = 0,
there is one
real solution.
solution.
0, there
•0 When
solutions.
real solutions.
0
are no
no real
there are
0, there
When D < 0,
• When
the vertex,
vertex,
find the
To find
get
to get
quadratic to
the quadratic
that value
Then plug
0
average of the
the roots
roots to get
get the
the x-coordinate.
x-coordinate. Then
plug that
value into
into the
the average
Take the
• Take
the y-coordinate.
y-coordinate.
the
square .
the quadratic
Put the
quadratic in vertex
vertex form
form by
by completing
completing the square.
•0 Put
2 is positive 1 by dividing everything by a.
coefficient of xx2
Ensure the
the coefficient
positive 1by dividing everything by
1. Ensure
C
b
2
y =x2+
x+
+x
-a = X + -E
a
a
b
b2
b
that result
Square that
2a. Square
get 2.
term bx to getZa
middle term
the middle
2. Divide
Divide the
the coefficient
coefficient of the
result to
to get
g e4a1,22
t 2‐ .. PPutu 2a
2a
t ‐
2.
!
2
end.
the end.
subtract £72 at the
with x and
inside
parentheses with
and subtract
the parentheses
inside the
z_(.+£)2+£_i
+:. - ~
(x+ _!:_)2
a 4a
2a
a
¥_=
a_
211
by a.
3.
everythingby
Multiply everything
3. Multiply
2
b 2
a
4:122
2
b2
c ‐ ‐b ‑
= a ( x +-b ) + c-yy=a(x+‐‐2)
4a
2a
actual
with actual
quadratics with
on quadratics
variables. Practice
the variables.
with the
4. IIt's
t ’s unnecessary
unnecessary to memorize
memorize these
these steps
steps with
Practice on
k.
+
)2
h
a(x
=
y
like:
looks
form
vertex
numbers.
However, do
do remember
remember what
what vertex form looks like: y : a(x‐- h)2 + k.
numbers. However,
a quadratic, find
minimum or
the minimum
asked for the
you're asked
Whenever
or the
the maximum
maximum of aquadratic,
find the
the vertex.
vertex.
Whenever you're
157
QUADRATICS
CHAPTER
17 QUADRATICS
CHAPTER 17
CHAPTER EXERCISE: Answers for this chapter start on page 314.
the
used on the
be used
NOT
should N
calculator should
A calculator
O T be
following
questions.
following questions.
3x2+10x=8
3x 2 + lOx = 8
equation
the equation
solutions to the
If
w o solutions
the ttwo
are the
and b are
If a and
?
b2 ?
value of If2
the value
above
what is the
> b, what
and a >
above and
the
between the
the distance
what is the
xy-plane, what
the xy-plane,
In the
distance between
parabola
the parabola
x-intercepts of the
ttwo
w o x-intercepts
yy = x2
10??
3x ‐- 10
x2 ‐- 3x
4
A) 9
2
B) 3
A) 3
B) 5
C) 4
C)
16
D) 16
D)
C) 7
D)) 110
D
0
fJ(x)
(x) =
=m[(x‐m)2‐l]
m [(x - m) 2 - 1]
to x 2 + 4x + 2 = 0 ?
solutions tox2+4x+2=0?
the solutions
What are the
Whatare
positive
a positive
m is a
above, m
defined above,
hmction ff defined
the function
In the
A
i fi
= z- 2‐ ±Z vf2
A) ) x x
B
n/i
B)) x x= 2z±e2v12
C) ) Xx=C
= ‐ 22 i±22y12
\/§
D
= ‐- 44 i±Z2,/2
\/§
D)) xx=
xy-plane is a
the xy-plane
offf in the
graph of
The graph
constant. The
constant.
following statements
the following
parabola.
statements
Whkh of the
parabola . Which
true?
parabola is true?
the parabola
about the
about
( m, ‐- 11).
at (m,
occurs at
minimum occurs
A) [Its
t s minimum
).
- m).
at (m, ‐m).
occurs at
minimum occurs
Its minimum
B) Its
B)
1).
(nz, ‐- 1).
at (m,
occurs at
C)
maximum occurs
Its maximum
C) Its
m).
(m, -‐ m
at (m,
occurs at
maximum occurs
Its maximum
D) Its
D)
).
= 0, what is the value
7a + 3 =O,whatisthevalue
2a 2 ‐- 7a+3
lIff a < 1
and 2a2
l and
o
a??
off a
yy = ‐- 33
2
yzx2+cx
y = x + ex
constant.
a constant.
c is a
above, 0is
equations above,
system of equations
In
the system
ln the
the
does the
values of c does
following values
the following
For
which of the
For which
real
exactly ttwo
have exactly
equations have
system
w o real
system of equations
solutions?
solutions?
solutions of
the solutions
the ssum
What
u m of the
What is the
2 =4x+5?
(2x‐3)2
(2x - 3) = 4x + 5?
A
A)) -‐ 4
B) 1
C) 2
D) 3
158
THE
PANDA
THE COLLEGE
COLLEGE PANDA
A calculator is allowed on the following
questions.
m 2 ‐- 100m -‐ 120,000
P=
= m2
120,000
The
The monthly
monthly profit
profit of a mattress
mattress company
company can
can
be modeled
modeled by
the equation
equation above,
above, where
Pisis
be
by the
where P
the profit,
profit, in dollars,
dollars, and
and m is the
number of
the
the number
mattresses sold.
the minimum
minimum number
number
mattresses
sold. What
What is the
of mattresses
sell in a
given
mattresses the
the company
company must
must sell
a given
month
so
that
it
does
not
lose
money
during
that
month so that does not lose money during that
month?
month?
At which
the following
following points
does the
which of the
points does
the line
line
the parabola
parabola
with
with equation
equation y =
z 4 intersect
intersect the
y = (x + 2)2
2)2 -‐ 55 in the
xy-plane?
the xy-plane?
A)
1,4 ) and
A) ((‐1,4)
and (( ‐ 55,, 4 )
B)
5,4 )
B) (1,4
(1,4)) and
and ((‐5,4)
C)
(1,4 ) and
4)
C) (1,4)
and (5,
(5,4)
D)
11, 4) and
D) ((‐11,4)
and (7,4
(7,4))
y
E(x)
= 50x
50x22 ‐- 8003:
+ 10,000
10,000
E(x) =
B00x +
The function
function above
above models
the relationship
relationship
The
models the
between the
the total
total monthly
expenses E, in dollars,
dollars,
between
monthly expenses
its
restaurant and
and the
the number
tables x in its
of a restaurant
number of tables
dining area,
area, where
where O::;
What does
does the
the
dining
O5 x::;
x 5 25. What
number 10,000 represent
the function?
function?
number
represent in the
(3, -8)
maximum number
tables that
A) The
The maximum
number of tables
that can
can fit
in the
the dining
dining area
area
Which of the following equation s represents the
parabola shown in the xy-plane above?
A) y
= (x - 3)2 -
8) y =(x+
C) y
= 2(x -
average monthly
expenses, in dollars,
dollars,
B) The
The average
monthly expenses,
for each
table in the
dining room
room
each table
the dining
C) The
dollars,,
The total
total monthly
monthly expenses,
expenses, in dollars
when there
there are
tables in the
dining
when
are zero
zero tables
the dining
8
3)2+ 8
area
area
3)2 - 8
D) The
The total
total monthly
monthl y expenses,
expenses , in dollars,
dollars,
when the
the number
number of tables
tables in the
the dining
dining
when
area
maximized
area is maximized
D) y= 2(x+ 3)2- 8
2
For
value oft
of t does
the equation
equation vv = St
St -‐ tt2
does the
For what
what value
result in the
maximum value
value of v ?
result
the maximum
f (x) = - x 2 + 6x + 20
The
defined above
above.. Which
the
The function
function f is defined
Which of the
following
forms of f (x)
displays the
(x ) displays
the
following equivalent
equivalent forms
maximum
value of f as
maximum value
as a
a constant
constant or coefficient?
coefficient?
A
( x )=: ‐-(( xx‐-3 3)2
) 2 ++ 11
A)) ff (x)
11
2
B
)
f
(
x
)
:
‐
(
x
‐
3
)
2
+
2
9
B) f (x) = -(x - 3) + 29
C)
f(x)=‐(x+3)2+11
C) f(x)
= - (x + 3)2 + 11
D)
D) ff(x):‐(x+3)2+z9
(x)=-(x+
3)2+ 29
159
CHAPTER 17
QUADRATICS
CHAPTER
17 QUADRATICS
yy =
= ‐- 33
g(x)
= - 3x 2 + 18x
The
gives the
the data
The function
function 3
g above
above gives
data transfer
transfer
speed,
second, over
network
speed, in megabytes
megabytes per
per second,
over a
a network
connection
connection x minutes
minutes after
after a file transfer
transfer was
was
initiated.
g(x)
in
the
xy-plane
initiated. The
The graph
graph of y =
= g(x) the xy-plane
has
c. Which
Which of
of the
has x-intercepts
x-intercepts at
at x = 0
0 and
and x = c.
the
following is the
following
the value
value
the best
best interpretation
interpretation of the
y=ax2+4x‐4
y = ax2 + 4x - 4
In the
system of equations
constant.
the system
equations above,
above, a
a is aa constant.
For
which of the
the following
For which
does the
following values
values of a
a does
the
system of equations
equations have
system
have exactly
exactly one
one real
real
solution?
solution?
cC
_7
?.
of 2
A)) -‐ 4
A
2
B)) -‐ 2
B
A) The
The initial
initial data
data transfer
transfer speed
speed over
over the
the
network connection
connection
network
C) 2
D)) 4
D
B)
B) The
speed over
over the
the
The maximum
maximum data
data transfer
transfer speed
network connection
connection
network
C) The
data transfer
The time
time at which
which the
the data
transfer speed
speed
over the
the network
network connection
over
connection was
was at its
highest
highest
f ( x ) = x 2‐ 24x +180
time at which
D) The
The time
which the
the file transfer
transfer
completed
completed
For a
a manufacturer
For
x‐ray machines
machines,, the
manufacturer of x-ray
the cost
per
unit,
in
thousands
per unit, thousands of dollars,
can be
dollars , can
be modeled
modeled
the function
function f above,
above, where
by the
where x is the
the weekly
weekly
number of units
units produced.
produced. How
number
How many
many units
units
should the
the manufacturer
manufacturer produce
produce each
should
week to
each week
minimize the
the cost
minimize
cost per
per unit?
unit?
y
= a(x -
3)(x - k)
the quadratic
In the
are
quadratic equation
equation above,
above, a
a and
and k are
constants. If
constants.
graph of the
If the
the graph
the equation
equation in the
the
xy-plane is a
xy-plane
(5, -‐32),
a parabola
parabola with
with vertex
vertex (5,
32), what
what
is the
the value
value of a ?
?
A) 2
B) 5
C) 6
D) 8
f(x) =
: -‐4x2
x
J(x)
4x 2 + 2
22x
The function
The
function ff above
above gives
gives the
the data
data transfer
transfer
speed,
in
megabytes per
per second,
speed, megabytes
second, over
over a
a network
network
connection x minutes
connection
transfer was
was
minutes after
after a
a file transfer
initiated.
graph of y = J(x)
f(x) in the
xy-plane
initiated . The
The graph
the xy-plane
has
x-intercepts at
= 0 and
= b. Which
has x-intercepts
at x =
and x =
Which of
of the
the
following is the
following
the best
best interpretation
interpretation of b
b??
In the
line y =
= 2x +
+ b intersects
the xy-plane,
xy-plane , the
the line
intersects the
the
parabola
at
the
point
(3,k).
If
parabola y =
= x2
x 2 + bx + 5
5 at the point (3, k). If b
b
is a constant,
is
the
constant, what
what the value
value of k ?
A) 0
A) The
the
The initial
initial data
data transfer
transfer speed
speed over
over the
network connection
network
connection
speed over
over the
the
B) The
The maximum
maximum data
data transfer
transfer speed
network
network connection
connection
C) The
The time
time at
at which
which the
the data
data transfer
transfer speed
speed
over
a s at its
its
over the
the network
network connection
connection w
was
highest
highest
The time
at which
D) The
time at
which the
the file transfer
transfer
completed
completed
B) 1
C) 2
D) 3
160
18
e
•
Division
Synthetic Division
Synthetic
way you
same way
the same
another in the
involves dividing
Synthetic
dividing one
one polynomial
polynomial by another
you divided
divided numbers
numbers in
in 3rd
3rd
division involves
Synthetic division
grade.
.
grade
xx 2 2++ 3x
3 x 7 22
1 8 R 2
3 5 6
3156
R
R ‐- 1]
+ 2x2 5x + 1
xv1|Fx3+2x2~5x+1
x - 1 Ix 3
several shortcuts
towards several
you towards
direct you
then direct
way first, but
"mathematical" way
I'll teach
l’ll
teach you
you the long
long “mathematical”
but then
shortcuts that
that will
will get
get
questions
way. These
long way.
without using
on the SAT without
you through
through almost
almost any
any synthetic
synthetic division
division question
question on
using the long
These questions
you
do, they'll
rarely
show up,
up, and
and if they do,
they’ll show
show up
up only
only once
once..
rarely show
synthetic division.
to synthetic
applies to
same logic applies
can see how
by 33 so
Let’s retrace
steps of dividing
dividing 56
56by
so you
you can
how the same
division.
retrace the steps
Let's
to get
and a 1 x 3 =
on top
once . We put
First, we see that
that 3 goes into
into 5 once.
put aa 1“1on
top and
z 3 below
below the 5. We then
then subtract
subtract to
get 22
First,
and
down.
bring the 6 dOWn.
and bring
l
6
3156
3
26
26
below the
24 below
into 26? 8 times
does 3 go into
times does
many times
N o w how
how many
times.. So we put
put an 8 up top and
and a 3 x 8 =
z 24
the 26.
26.
Now
Subtracting,
Subtracting, we get 2.
l 8
3156
3
2 6
2 4
- 2-
eighteen
into 2. Therefore
does nnot
and 3 does
At this point,
point, there
there are
are no more
m o r e digit
digitss to bring
bring down
down and
o t go into
Therefore,, 3 goes iinto
n t o 56
56 eighteen
At
form:
following
the
in
written
be
can
result
This
two
times
w
i
t
h
a
remainder
of
t
w
o
.
result
can
be
written
following
remainder
with
times
56 = 18~
3
3
divisor .
and 3 is the divisor.
quotient, 2 is the remainder
where 18
18is
the quotient,
remainder,, and
is the
where
161
CHAPTER 18
18 SYNTHETIC
SYNTHETIC DIVISION
CHAPTER
DIVISION
The process
The
essentially the
the same.
same. To
show you
you how
synthetic division
division works,
process of dividing
dividing a
a polynomial
polynomial is essentially
To show
how synthetic
works, let’s
let's
3 + 2x
divide
divide xx3+
2x22 -‐ Sx
5x + ll by
by x -‐ l.1.
3? xx22 times.
3. We
H
o w many
many times
We
How
times does
does x -‐ 1
l go into
into xx3?
times. Why?
Why? Because
Because xx xx x2
x2 =
= x3.
x3. The
The goal
goal is
is to
to match
match xx3.
3- x
2 . This
don’t
i n " step.
step. Now,
Now, (x
1) xx x2
x2.
what we
we put
don ' t care
care about
about the
the -‐11 during
during this
this ”fitting
"fitting in"
(x ‐- 1)
x2 = xx3‐
This is
is what
put below
below
dividend.
the dividend.
xx22
3
xx -‐ l1x
j x3
++ 22xx2 2- ‐ 5Sxx ++ 11
xx33 ‐ x x2
2
Finally, we
subtract like we
Finally,
we subtract
do in basic
number division.
division. Notice
Notice that
subtract each
we do
basic number
that we
we must
must subtract
each element,
element, sothe
so the -~x2
x2
2
2
becomes
x , yielding
becomes +
+x2,
yielding 3x2.
Unlike in long
long division
division with
dividend
3x . Unlike
with numbers,
numbers, all the remaining
remaining terms
terms from the
the dividend
should be
be brought
brought down
down for each
should
each step
step in synthetic
synthetic division.
division.
x2
x2
xx33
xX -‐ l11 ++ 22xx22- ‐ 5Sxx ++ 11
x3
x 3 ‐ x x2
2
3 x2 2- ‐ Sx
5 x++ 11
3x
2 ? 3x
Next step.
step. H
How
many times
time s. Remember
Next
o w many
times does
does x -‐ 1 go into
into 3x
3x2?
3x times.
o u r goal
Remember our
goal at each
each step
step is to get
the same
same exponent
exponent and
and the
the same
same coefficient
the term
with the highest
the
coefficient as
as the
term with
+3x up
and
highest power.
power . We put
put the
the +3x
up top
top and
3x x (x -‐ 1) = 3x2
3x 2 -‐ 3x
3x on
on the
the bottom.
bottom .
3x
x2
x 2++ 33xx
+ 22xx2 2- ‐ 5
+ 11
Sx x +
xx -‐ 1
l jxx33 +
xx33 -‐ x x2
2
3 x2 2 ‐ Sx
5 x++ 11
3x
3 x2 2 ‐ 33xx
3x
And just like last
last time,
subtract each
each term,
term, nnot
And
time, we subtract
o t just
just the
1.
the first. We then
then bring
bring down
down the
the 1.
x2
x 2 ++ 33xx
x ‐ 111
xx33 ++ 22xx2 2- ‐ 5
+ 11
Sx x +
x3
x 3 ‐ xx2
2
3 x2 2- ‐ Sx
5 x++ 11
3x
2
- 3x
3x2‐3x
3x
-‐ 22xx +
+ 1
1
X -
162
THE COLLEGE
COLLEGE PANDA
THE
PANDA
We’re
How many
many times
does xx -‐ l1 go
go into
1? -‐22 times
times.. So
up top
and
We're almost
almost done.
done. How
times does
into -‐ 22xx +
+ 1?
So a
a ‐2
- 2 goes
goes up
top and
1) =
= -‐ 22xx + 2 goes
the bottom.
bottom.
-‐22 x (x -‐ 1)
goes on
on the
x2
x 2+ + 3x
3
x ‐ 22
x33 ++ 22xx2 2 ‐ 5
5xx ++ 1
xx -‐ 1
l lx
1
‘%
xr :1
‐
x2
x2
2 2 ‐ 5x
3xx
3
5 x++ 11
3x 2
3x2‐3x
3x
‐ 22xx ++ 1
1
‐ 22xx ++ 2
Subtracting, we
Subtracting,
we get
get ‐1
- 1 at
at the
the end.
end .
x2
X -
+ 3x
3
1 1x + 2x 2
x3
3 : 3 ‐ xx2
2
2
5x +
1
2 2 ‐5x
3x x
3
5 x+ + 11
3x 2
3x2‐3x
3x
‐ 22xx ++ 1
‐ 22xx ++ 2
-‐11
.
.
56
2
.
.
56
know we're
we’re done
With a constant.
constant. And
185,
m1xed fraction,
fraction,
We know
done when
when we end
end up with
And Just
just as
as we can
can express
express ‐3‐ as
as 18~,
aa mixed
3
we
we can
can express
express
x33++22xx 22_- 5x
5 x+
+ 1 as x2+3x‐2‐
2
11
as X + 3x - 2 - -x -‐ 1l
x -‐ 1l
Notice where
Notice
each component
component is placed.
placed. The
The quotient
u t in
remainder, -‐ 11,, is
is the
the
where each
quotient is written
written oout
in front.
front. The
The remainder,
numerator of the
numerator
fraction
and
the
divisor,
x
‐
1,
is
the
denominator.
These
placements
are
exactly
the
same
as
the fraction and the divisor, - l, is the denominator. These placements are exactly the same as
long division
in long
Get used
used to
to seeing
seeing synthetic
results in
in this
division with
with actual
actual numbers.
numbers . Get
synthetic division
division results
this format.
format.
Here’s another
that’s the
the same.
same. The
The result
result of
of our
o u r long
long division
numbers
Here's
another thing
thing that's
division with
with numbers
1 8 R 2
3 5 6
3156
means
56 =
18 + 2.
means that
that 56
= 3 x 18
The same
meaning applies
our synthetic
synthetic division
The
same meaning
applies to
to our
division result.
result .
x3+2x2‐5x+1=(x‐1)(x2+3x‐2)‐1
x 3 + 2x 2 - 5x + 1 = (x - l ) (x 2 + 3x - 2) - 1
Dividend =
= Quotient
Quotient x Divisor
Remainder
Dividend
Divisor + Remainder
Hopefully you've
you've been
to grasp
synthetic division
o r e intuitively
Hopefully
been able
able to
grasp synthetic
division m
more
intuitively through
through the
the comparison
comparison with
with regular
regular
long
the parts
each other
other in
in the
where
long division.
division. A
Alll l the
parts relate
relate to each
the same
same way.
way. Let's
Let's dive
dive into
into some
some more
more examples
examples where
we
show you
we can
can show
you some
some shortcuts.
shortcuts.
163
CHAPTER
CHAPTER 18
18 SYNTHETIC
SYNTHETIC DIVISION
DIVISION
6
5
EXAMPLE
expression (2:25
x - is equivalent
the following?
following?
EXAMPLE 1: The
The expression
equivalent to which
which of the
x+ 2
17
A’6“m
x+2
A)6 -~
B)6+
7
_2_
B)6+x+2
x+2
6 -‐ 5
C) 6-5
07
5
0)6 - -
9’6‘52
2
Using
division,
Using synthetic
synthetic division,
6
x
X
+ 22 I6x
6x
+
‐
5
6x +
+ 12
6x
‐
17
17
The quotient
quotient is 6 and
and the
the remainder
remainder is -‐ l17.
7 . We
We can
can write
write this
this result
xl‐ZZ'-An
Answer
The
result as6
as 6 -‐ ~
swer~-(A) .
x +2
~
Now
question without
division?
N o w how
how would
would we
we approach
approach this
this question
without using
using synthetic
synthetic division?
6xx-‐ 5
6(2)
77
.
6
6
( 2 )- ‐ 5
We
up.. Let's
say x == 22.. ' IThen
w e ccan
a nplug
p1u g in
l n numbers
n u mb ers that
t hat we
w e make
m ak eup
Let ’ssayx
' he n --x + 2 = ---2 + 2 = ‐-4 .
x+2
2+ 2
4
7
n o w look
look for an
an answer
answer choice
choice that
that gives
gives 2 when
when x =
z 2. We can
rule out
right away
away since
We now
can rule
out (C) and
and (D) right
since they
they
4
i·
don't give
give 2. Plugging
Plugging x =
z 2 into
into answer
answer (A) gives
gives
don't
_17_617
x+2 _
4
2417
4
4
7
4
This
making up
numbers and
testing each
each answer
This confirms
confirms that
that the
the answer
answer is indeed
indeed (A)
(A).. This
This strategy
strategy of making
up numbers
and testing
answer
choice
be much
choice can
can be
much faster
faster than
than synthetic
synthetic division.
division.
7
EXAMPLE
2: When 3x 2 + 4 is divided by x - 1, the result is A + -. What is A in terms of x ?
EXAMPLBthensz+4isdividedbyx‐1,theresultisA+;‐_7:‐1.WhatisAintermsofx?
x- 1
A)3x
A ) 3 x-‐ 4
B)3x-3
B)3x‐3
C)3x
C ) 3 x++ 3
D)3x+4
D)3x+4
Using synthetic
synthetic division,
division,
Using
3x + 3
x
X
3x22 + 4
-‐ 1 I3x
3x22 -‐ 3x
3x
3x + 4
3x ‐ 3
7
If you
followed along,
clunky when
when we subtracted
subtracted the
the -‐ 33xx and
and brought
brought
you followed
along, you
you should've
should’ve noticed
noticed it got
got aa little
little clunky
the 4 down.
That’s because
because the
the dividend,
dividend, 3x
3x22 +
+ 4,
4, has
has no
no x term.
process is the
the
down. That's
term. Still, the
the process
the same:
same: subtract
subtract and
and
bring the
remaining terms
terms down.
down.
bring
the remaining
164
THE
COLLEGE PANDA
PANDA
THE COLLEGE
L1.
2
4 : 3x + + 7
The quotient
quotient is 3x
3x +
+ 3 and
the remainder
7. The
The result
result can
as 33x
+14
Now
x +
The
and the
remainder is 7.
can be
be expressed
expressed as
= 3x + 3 + _ __ Now
x ‐- 1
x -‑ 1
it’s easy
easy to see
see that
that A =
= 3x
3x +
+ 3,
3, answer
answer ~-(C) .
it's
Again, we
we could've
could’ve done
done thi
thiss question
question by
by making
making up
up numbers.
lf x =
2, then
Again,
= 2,
then
numbers. If
2
3x22 +
+44 _ 3(2)
3(2)2+4
3x
+4
16
- xx---‐ 1
1- = - 2
2---‐ 1
1- =
216
If we
we didn't
didn’t know
know the
the answer
answer was
was (C),
(C), we
we would
would test
test each
answer choice
with xx = 22 until
16, but
since
If
each answer
choice with
until we
we got
got 16,
but since
we do
do know,
know, we'll
we’ll test
(C) first for confirmation.
confirmation. Letting
Letting A =
= 3x
3x +
+ 3,
we
test (C)
3,
7
7
3X+3+m
‐3(2)+3+m
~ ‐99++77=: 116
6
+ 3 + -=
3(2) + 3 + -=
x- 1
2- 1
3x
Answer
Answer confirmed.
confirmed.
2
4
1
EXAMPLE
3: If the expression Sx - ~ + is written in the form Sx + 6 + __!!_, where B is a constant,
EXAMPLEB:Hfl\eexpressim-5’:2‐Jt_‐__4ii‐liswritteniritheform5x+6+‐B-,whereBisaconstant,
xx-‐ 2
x- 2
what
is the value
off B ?
whatisthe
value o
Based
division.
Based on
on where
where it is,
is, B represents
represents the
the remainder
remainder of the
the division.
5x
5 x+
+
x -‐
X
6
2 I5x
5x2‐ 4x
4x
2
+1
+ 1
2
5x2‐10x
5x
lOx
6
6x x ++ 1
6x ‐
12
13
13
13
r:;---:;-i
Wecan
can write
write the
the result
result of this
this division
division as
as5x
x _ 2,, from
from which
= L.llJ.
-.
We
Sx + 6 + -which B =
2
xThis last
last example
example is perfect
perfect for demonstrating
demonstrating aa shortcut
shortcut called
remainder theorem,
get
This
called the remainder
theorem, which
which allows
allows us
us to get
the
remainder without
without going
going through
through synthetic
synthetic division.
division.
the remainder
In Example
Example 3,
3, we
we divided
divided Sx
5x22 -‐ 4x
4x +
+ 11by
2. Whenever
Whenever a
divided by
by a
which is
is
1n
by xx -‐ 2.
a polynomial
polynomial is divided
a monomial,
monomial, which
just something
something in the
the form
form of ax
ax + b,
b, the
remainder can
can be
be found
the va
value
the remainder
found by
by plugging
plugging in to the
the polynomial
polynomial the
lue
just
that makes
makes the
the monomial
monomial equal
equal to
to 0. The
The process
process sounds
m o r e complicated
than it is,
so let’s
of xx that
sounds more
complicated than
is, so
let's show
show how
how
it’s done
done..
it's
What makes
0? x =
z 2.
What
makes x -‐ 2 equal
equal to O?
Plug xx = 22 into
the polynomial
polynomial 5x
5x22 -‐ 4x
4x + 1.
1.
Plug
into the
5(2)22 -‐ 4(2) +
+11 = 13
13
5(2)
A
n d that's
that’s the
the remainder
remainder we
we obtained
obtained in Example
Example 3.
And
165
DIVISION
SYNTHETIC DIVISION
18 SYNTHETIC
CHAPTER 18
CHAPTER
l?
+ 1?
divided by x +
- 2x 2 +
when ‐23(2
remainder when
the remainder
What is the
What
+ 5x is divided
polynomial,
the polynomial,
into the
Plugging that
zero? x = -‐ 11.. Plugging
to zero?
equal to
makes x +
what makes
Well, what
+ l1equal
that into
7
‐2(‐1)2+5(‐1)
- 2(- 1) 2 + 5(- 1) = -‐7
the remainder
- 7 is the
Boom. ‐7
Boom.
remainder..
l?
divided by 2x
4x 4 +
when 4x4
remainder when
What
+ 3x
3x22 -‐ 4 is divided
2x -‐ 1?
the remainder
What is the
1
zero? x =
equal to zero?
makes 2x -‐ 1 equal
What makes
What
z 2'.
2
polynomial,
the polynomial,
into the
Plugging
that into
Plugging that
2 - 4 = 41+ 43- 4 = - 3
(21) + 3 (1)
4
4
2
remainder.
the remainder.
Boom . -‐33 is the
Boom.
.· 2x2‐5x+1.
.· m
.· alent form 2x + 1 + -x-R
R
.
·. th e equiv
2x2x- _Sx + 1 IS· wntten
uv"' 11.•PLE If th e expression
__- , what is the
EXAMPLEdzlftheexpreesmn‐‐;‐:§‐‐1swnttenmtheeqmvalentform2x+l+m,whatrsfire
3
3
J;,.IUUT-1,
A
,,.: .
value of R?
valueofR?
plug in
can plug
we can
theorem, we
remainder theorem,
the remainder
Using the
by x -‐ 3. Using
R represents
after dividing
dividing 2x
2x22 -‐ 5x + 11by
remainder after
the remainder
represents the
2
remainder.
the
get the remainder.
x = 3 into
+ 1 to get
2x -‐ 5x +
into 2x2
2(3)2‐5(3)+1=18‐15+1=.
2(3) 2 - 5(3) + 1 = 18 - 15 + 1 = [±J
division .
synthetic division.
No
need for synthetic
No need
Plugging in 2,
2. Plugging
x 2 ‐- 3x + 2 by x ‐- 2.
divide x2
we divide
that we
theorem . Let's
remainder theorem.
the remainder
about the
thing about
One
Let’s say
say that
2, we
we
last thing
One last
see
remainder is
the remainder
that the
see that
(2)22 -‐ 33(2)
+2
=0
2z
(2) +
(2)
factor
we factor
indeed, if we
factor of 18. And
a factor
like 3 is a
x 2 -‐ 3x + 2, just
remainder is 0, x ‐- 2 is a factor of x2
the remainder
Since
just like
And indeed,
Since the
2 ‐ 3x + 2,
x2
2,
+
3x
x 1)
2)( x ‐- 1)
(x -‐ 2)(X
(x
that x -‐ 2 is in fact a factor.
we see
see that
connected?
math is connected?
everything in math
how everything
Don’t
love how
just love
you just
Don't you
1?
x 3 + 1?
factor of x3
1 a factor
+ 1a
example, is x +
easier . For example,
much easier.
become much
Some
o w become
questions nnow
Some questions
3+
factor of xx3
Therefore, x + 1 is a factor
= 0. Therefore,
- 1) 3 + 1 =
remainder is ((‐1)3
that the remainder
Well, plugging
1.
plugging in -‐ 11,, we find that
were
we were
If we
like x + 1.
monomials like
dividing by monomials
works when
only works
theorem only
remainder theorem
the remainder
that the
note that
Do note
when we're
we’re dividing
1. If
division .
synthetic division.
use synthetic
have to use
would have
we would
x2 + 2, we
dividing
like x2
something like
by something
+ 11by
x3 +
dividing x3
166
PANDA
COLLEGE PANDA
THE COLLEGE
THE
EXAMPLES:
EXAMPLE
5:
+5Sxx++ 2
kx2 +
3x 3 ‐- kx2
f(x)
= 3x3
f(x) =
k??
what is the
2, what
by x ‐- 2,
divisible by
constant. If ff(x)
a constant.
is a
above, kit is
defined above,
polynomial f(x)
the polynomialf
In ·the
(x) defined
(x) is divisible
the value
value of k
12
A) 12
B)9
B) 9
C)66
C)
0)33
D)
a
words, x ‐- 2 is a
other words,
2. In other
divided by x ‐- 2.
when ff(x)
O when
remainder is 0
the remainder
then the
2, then
If
f (x) is divisible
(x) is divided
divisible by x -‐ 2,
If f(x)
zero)
equal to zero)
makes x ‐- 2 equal
that makes
va lue that
(the value
plug 2 (the
we plug
that when
remainder theorem
factor
r ) The
theorem tells us
us that
when we
The remainder
factor of f ((x).
get 0.
into
(x),
should get
we should
), we
into fJ(x
f/ ((2)
2 ) = 00
3(2)33
3(2)
‐k(2)2+5(2)
0
+ 2 := o
k(2) 2 + 5(2) +2
2 4- ‐ 4
1 0++22=: 0
4kk++ 10
24
-
36 -‐ 4k =
36
=0
‐- 44kk := ‐- 336
6
k= 9
Answer
Answer ~(B) .
EXAMPLE
EXAMPLE 6:
X
p(x)
-3
1
- 1
0
0
5
2
-3
4
4
the following
Which of the
values of x. Which
some values
p(x) for some
polynomial p(x)
value of polynomial
the value
gives the
above gives
table above
The table
The
following must
must be
be
p(x)??
a factor of p(x)
A
) x ++ l1
A)x
B)x
B ) x- ‐ 1
C)x
C
) x-‐ 4
D)x
D ) x ‐- ‐ 5
Answer
p(x). Answer
factor of p(x).
a factor
be a
must be
= 0, x ++ 1 must
p((-‐ 1)
make s this
theorem makes
remainder theorem
The remainder
The
this question
question easy.
easy. Because
Because p
l) =
is NOT
answer is
Be careful-the answer
NOT x -‐‐ 1.
1.
~-(A) . Becareful‐the
167
CHAPTER
CHAPTER 18
18 SYNTHETIC
SYNTHETIC DNISION
DIVISION
CHAPTER EXERCISE:Answers for this chapter start on page 317.
A calculator should
be used
used on the
should NOT
N O T be
the
following
following questions.
questions.
X
The
expression x4_x
x 2 is equa
The expression
equall to which
which of the
the
x- 2
following?
following?
3
6
0
A)
A ) -‐ 2
8
B) ---+
x- 2
1
2
3
-4
-3
-3
-2
4
4
g(x)
The
function g is
is defined
defined by
by a
a polynomial.
polynomial. The
The
The function
g(x).
table above
above shows
shows some
va lu es of x and
some values
and g(x).
table
What is the
the remainder
remainder when
when g(x)
g(x) is divided
divided by
What
by
xX + 33??
8
4
) x -‐ 2 +
- 2x
2x
D) 44 ‐‐
C) --+
C
‐‐
A)) -‐ 2
A
B) 1
C) 2
2
.
. th
. 6x + 5x + 2 .
IIff the
expression M x +
1s
th e expression
is wntten
written m
in thee
22x + 11
__
' '
D) 6
7
1
Q, what is Q in terms of x ?
form _2 xl_+ 1 +
form
+Q,whatisQ1ntermsofx.
2x + 1
A)
A ) 33x
x -- 1
2z3 - kxz2 + 5xz + 2x - 2
223‐kxzz+5xz+2x‐2
B) 3x + 1
C)
C) 6x2+3x+1
6x 2 + 3x + 1
In the
the polynomial
polynomial above,
is aa constant.
constan t. If 2
z -‐ 11
above, kk is
is a
a factor
factor of the
the polynomial
polynomial above,
what is
the
is
above, what
is the
D)
D) 6x2+5x+1
6x 2 + 5x + 1
value
value of kk??
The
written as
The expression
expression 4x
4x22 + 55 can
can be
be written
as
A(2x
1)
+
R,
where
A
is
an
expression
A(2x ‐
R, where A anexpression in terms
terms
of x and
lu e of R
and R
R is aa constant.
constant. What
What is the
the va
value
R ??
What is the
th e remainder
remainder when
x 2 + 2x
2x + 1 is
What
when x2
is
divided
divided by
by xx ++44 ??
168
THE
PAN DA
THE COLLEGE
COLLEGE PANDA
A calculato
calculatorr is allowed
allowed on the following
following
question s.
questions.
3 + x2
If p(x)
= xx3
divisible
p(x) =
x2 ‐- 5x + 3,
3, then
then p(x)
p(x) is
is divisible
by which
which of the
the following?
following?
by
I. xX -‐ 2
When
Bx -‐ 4 is
by 3x -‐ 2,
When 3x
3x22 -‐ 8x
is divided
divided by
2, the
the
result
can be
be expressed
A -‐
result can
expressed as
asA
II. x ‐- l1
II.
IIII.
I]. x +
+ 33
x ~ . What
What is A
£133
A
2
3
A)
A ) xx-‐ 4
and II
only
A) I and
11only
and 11]
III only
only
B) I and
C) [I
II and
and IIII
only
C)
l l only
B)) xx-‐ 2
B
D) I,
I, II,
II, and
and III
D)
III
in
terms of x ?
intermsofx?
C)
C ) xx ++ 2
D)
D ) xx ++ 4
the polynomial
polynomia l p(x)
p(x) is
is divisible
divisible by
by x ‐- 2,
If the
2,
p(x)??
the following
could be
be p(x)
which of the
which
following could
The
The expression
expression 2x
2x22 -‐ 4x
4x -‐ 33 can
can be
be written
written as
as
A
(x +
where Bis
constant. What
What is A
A(x
+ l1)) +
+ B, where
B is a constant.
in terms
terms of x ?
= - x 2 + 5x - 14
B) p(x)
= x 2 - 6x - 2
B)
p(x)=x2~6x‐2
C) p(x)
= 2x 2 + x - 8
C)
p(x)=zx2+x‐8
D) p(x)=
3x2 - 2x - 8
D)
p(x)=3x2‐2x‐-8
A) p(x))
A) 2x
A)
2x +
+6
B) 2x
2x + 2
C)
C ) 22xx -‐ 2
D)
D ) 22xx -‐ 66
If x ‐- 1 and
and x +
are both
both factors
factors of the
the
If
+ 1 are
2
4
3
po lynomial ax
bx -‐ 3x
and a and
and bare
polynomial
ax4 + bx3
3x2 + 5x
5x and
b are
constants, what
what is the
the value
value of a ?7
constants,
The
can be
The expression
expression xx22 + 4x
4x -‐ 9 can
be written
written as
as
(ax+
b, and
and care
(ax + b)(x
b)(x -‐ 2) +
+ c, where
where a,
a,b,
c are
constants. What
constants.
What is the
the value
value of a +
+ b+
+ c?
A ) -‐ 3
A)
B) 1
C) 3
A ) -‐ 2
A)
D
D)) 5
B) 3
C) 7
D
D)) 1100
For a
For
a polynomial
polynomial p(x),
p(x), p
(5)
of the
(D= 0.0. Which
Which of
the
following
following must
must be
be a
a factor
factor of p(x)
p(x)??
A) 3x ‐- 1
l
(2) = 0.
For a
a polynomial
polynomial p(x)
For
p(x),, pp(2)
0. Which
Which of
of the
the
folJowing must
be true
following
m u s t be
true about
about p(x)
p(x) ??
B
+ 1l
B)) 3
3xx +
A) 2x is a
a factor
factor of p(x).
p(x).
C) x - 3
p(x).
B) 2x -‐ 2 is aa factor
factor of p(x).
D) x + 3
C) x -‐ 2 is a
a factor
factor of p(x).
p(x).
+ 2 is aa factor
factor of p(x).
p(x).
D) x +
169
19
Numbers
ComplexNumbers
Complex
2 =
imaginary
the imaginary
invented the
mathematicians invented
until mathematicians
What
value of x satisfies
: -v1?
There were
w e r e no
no values
values until
1? There
satisfies xx2
What value
can
power
other
from there,
and from
They defined
;=1. They
represents \/‐1.
number
defined i2
i2 to equal
equal -‐ 11,, and
there, any
any other power of i can
which represents
number i, which
be derived.
be
derived.
;2
1 “ =: -7 1]
;3 =4-‐1i
1K
;4
14=
z 11
1'5 L
i
;6
1"= :- fl1
;7
{7 =z -‐1i
;B
i” =
z 11
powers of i. For
higher powers
simplify higher
i4 =
that i4
the fact that
repeat in cycles
The
cycles of 4.
4. You can
can use
use the
: 1l to simplify
For
results repeat
The results
example,
example,
1-511: (ml: X 1'2 ; 1 x i 2 : _]
subtract,
add, subtract,
We add,
complex number.
a complex
called a
like 3 +
When
used in an
an expression
expression like
+ 2i, the
the expression
expression is called
number. We
When i is used
expressions.
algebraic
would
we
like
much
numbers
complex
divide complex numbers much like we would algebraic expressions.
and divide
multiply,
multiply , and
equivalent to (3
the following
EXAMPLE 1: If i = J=I,
\/ ‐ 1 , which
which of the
following is
is equivalent
(3 +
+ Si) ‐- (2
(2 -‐- 3i)
3i) ??
EXAMPLE
A)9i
A)9i
B )11+
+2
2ii
B)
C
C)) 1 + 8Bii
D)
D ) 55 ++ 8Bii
Just expand
and combine
terms .
like terms.
combine like
expand and
Just
( 3+
+ Si)
5 1 )- ‐ (2
( 2 -‐ 33ii ) =
2 .3 +
+ 5Sii -- 2 + 3
: 11 ++8Bii
3i i =
(3
Answer
Answer ~(C ) .
170
THE
COLLEGE PANDA
PANDA
THE COLLEGE
A,
EXAMPLE
2: Given
Given thati
that i =
what is the product
(4 + ii))(5
EXAMPLEZ:
= \/‐1,whatisthe
product (4+
( 5- ‐ 2i) ?
A) 18 - 3i
A)18‐-3i
B)
22 - 3i
B)22‐-3i
C)
C ) 18
1 8+
+ 33ii -.
D)
D ) 22
2 2+
+ 3i
3i
Expanding,
Expanding,
((4+i)(5‐2i):20‐8i+5i‐2i2=20‐3i+2=22‐3i
4 + i) (5 - 2i) = 20 - Bi + 5i - 2i2 = 20 - 3i + 2 = 22 - 3i
Answer ~( B ) .
Answer
.
.
.
22 + 3 i
1 +i
EXAMPLE
the following
is equal to
~i ??
EXAMPLE 3: Which
Winch of the
followmgisequal
to 1+
+1.
1 1.
A)§‐§l
1..
1 1
B)
B)§+'il
-+- 1
2
2
5 1.
(Di‐El
55 1.
1.
D)
D)§+El
-+
-t
2 2
When you're
with aa fraction
containing i in the
denominator, multiply
multiply both
and the
the bottom
When
you’re faced with
fraction containing
the denominator,
both the top
top and
bottom of
the conjugate,
conjugate, you
conjugate of 1 + i
the
the fraction
fraction by the
the conjugate
conjugate of the
the denominator
denominator.. What
What is the
you ask? Well, the
the conjugate
simply reverse
reverse the sign
between.
is 1 -‐ i. The conjugate
conjugate of 5 -‐ 4i is 5 + 4i.
41". To get
get the
the conjugate,
conjugate, simply
sign in between.
In this
top and
this example,
example, we multiply
multiply the top
and the
the bottom
bottom by the
the conjugate
conjugate 1 -~ i.
2 + i - 3i2
5 +i
5 +1 i.
(2+3i).(1‐i)_2‐2i+3i‐3i2_2+i‐3i2_5+i
1
(2 + 3i) (1 - i) _ 2 - 2i + 3i - 3i2
(1
11-‐ i + i ‐- i ;2
( 1+
+ ii)) . (1
( 1-‐ ii)) 2
_
11‐12
- i2
= - 22- = 2 + 2
The
The whole
whole point
point of this process
process is to eliminate
eliminate i from the denominator
denominator.. The absence
absence of i in the
the denominator
denominator is
aa good
were done
answer is ~(D) -.
good indicator
indicator that
that things
things were
done correctly. The
The answer
171
CHAPTER 19
CHAPTER
19 COMPLEX
COMPLEX NUMBERS
NUMBERS
CHAPTER EXERCISE: Answers for this chapter star t on page 319.
A calculator
calculator should
should NOT
N O T be
used on
on the
the
be used
following questions.
following
questions .
((66++2i)(
fl fl 22+
+ Si)
5)
If
+ bi,
bi,
If the
the expression
expression above
above is
is equivalent
equivalent to
to aa +
where
a
and
b
are
constants,
what
is
the
value
where and bare consta nt s, what is the va lue
of aa??
For
z \/‐_1,
of the
following is
For ii =
;=T , which
whic h of
the following
is
equivalent
Si)??
equivalent to (5 -‐ 3i)
3i) -‐ (( -‐ 22 +
+ Si)
A)) 33 -‐ 8
Bii
A
A) 2
B) 3 +
+ 2i
B) 12
12
B)
C)
C) 7 -‐ Bi
81'
C)) 2222
C
D)
+ 2i
2i
D) 77 +
D)) 3344
D
Which
Which of the
following is
equall to
the following
is equa
to
Given that
that i = \/‐_1,
Given
of the
the following
following is
is
;=T, which
which of
equal to ii(i
( i + 1)
1) ??
3(i0++2)
3
m-‐ 2(5
2 w- - 4i)
4 0?u(Note:
w m ei =i =H¢ f fl)
A) i -‐ 2
A) 16
16 -‐ Si
B) i -‐ 1
B) -~44 + 7i
C) i +
+ 1l
C) -‐ 44 ++ 11i
Hi
D)
D) 0
D) 16
16 ++ 11i
11i
For i =
= \/R‐ l ,, which
For
is
which of the
the following
following is
equivalent
+
2)
‐
i
(
i
‐
1)
?
equ iva lent to 3i(i
3i(i + 2) - i(i - 1) ?
fi+3fl+2
i4 + 3;2 + 2
Which of the
the following
Which
following is equal
equa l to the
the expression
expression
above? (Note
above?
(Note:: i = H
\/‐_1))
A
A) ) ‐- 4 ++ 7ifi
A) i
A)
B
B) ) -‐ 22 ++ 7ifi
B
B)) -‐ 1
C
C)) -‐ 4 ++ 5ia
D
&
D)) ‐- 2 ++ Si
C) ) 0
C
D) 1
For i = \/
R‐ 1 , which
which of
of the
the following
following is
is equal
equal to
to
,-93 ?
?
193
22+m+43+5fi+6fl
+ 3i + 4i + 5{'- + 6i
2
4
A
A)) -‐ 1
If the
the expression
expression above
If
above is equivalent
to aa +
+ bi,
bi,
equiva lent to
where a and
where
value of
and b are
are constants,
constants, what
what is
is the
the value
of
a+
+bb?? ((Note
N o t ei i=
= \/‐_1)
H )
B
B) ) 1
C)) -‐ ii
C
D
D)) i
A)) 2
A
B) 6
C
0
C)) 110
D)) 112
D
2
172
THE COLLEG
COLLEGE
THE
E PANDA
PANDA
Which of the
following complex
numbers is
is
Which
the following
complex numbers
equivalent
(3
‐
i
)2?
(Note:
i
=
\/-‐‐1)
equivalent to
to (3 - i)2? (Note: i
J=T)
A)
A)
Which of the following is equal to -1 -- 3i. ??
3+ 1
(Note: i =
(Note:
= \/‐‐1)
R )
88 -‐ 61"
6i
B)
8) 88 + 61'
6i
A
A)) -‐ i ,-
C) 10
10 -‐ 6i
C)
6i
B)
8) i
+
C)‐§i
C) - ~i
+
D)
0)
10
10 + 61’
6i
0) ‐‐§i
~ - ~i
D)
4 4
(-i)2
- i)4
(‐i)2 -‐ ((‐i)4
number system,
system, what
what is
is the
In the
the complex
complex number
the
value
(Note:
= \/‐_1)
value of the
the expression
expression above?
above? (Note: ii =
J=T)
Which
the following
Which of
of the
following complex
complex numbers
numbers is
is
2
equivalent to 3‐1;
equivalent
: ‘/_1)
- ~ ?? (Note:
(Note: ii =
J=T)
A)) -‐ 2
A
B)
8) 0
2+ 1
C) 1
3 4,
- - l
2‐31“
5 5
4,
8) 1 - - I
5
5 4.
C) - - - l
3 3
4,
0) 1 - - /
3
A)
A)
D) 2
0)
-
B)1‐§i
C)g‐gi
(5
(5 -‐ 2i)(
2i)(4
31')
4 -‐ 3i)
D)l‐§i
Which of the
the following
following is
is equal
equal to
to the
the expression
Which
expression
above?
above? (Note:
(Note: ii =
= \/‐_1)
R )
_I__________
__,
A
4 -‐ 7
A)) 114
7ii
B)
14 -‐ 23i
8) 14
23i
C)
+
7i
C) 26
26
4 + i + 2 ‐1
- i
1 ‐1
- i+ l +
+1i
D) 26
0)
26 ‐- 231"
23i
Which
to the
Which of the
the following
following is
is equal
equal to
the expression
expression
above?
= \/‐‐1)
above? (Note:
(Note: ii =
J=T)
A) -‐22 -‐ i
B) 2 + i
8)
1
11
1
Which of the following is equal to -:i‐2+i_4'
+~
+~ ?
I
(Note: i
I
I
C) 4 + i
= J=T)
D)
0) 4 ‐- i
A) - i
8)
C) 0
0)
1
173
20
Absolute Value
Absolute
The absolute
absolute value
value of x, denoted
denoted by lxl,
|x|, is the
the distance
distance xis
x is from
from 0.
O. In other
other words,
words, absolute
absolute value
value makes
makes
The
everything
everything positive.
positive. If it's
it’s positive,
positive, it stays
stays positive.
positive. If it's
it’s negative,
negative, it becomes
becomes positive.
positive.
EXAMPLE 1: How
H o w many
many integer
integer values
values of x satisfy
satisfy lxl
|x| << 44??
EXAMPLE
Think
the negative
Think of the
the possible
possible numbers
numbers that
that work
work and
and don't
don’t forget
forget the
negative possibilities.
possibilities. Every
Every integer
integer between
between -‐33
and
and 33 works,
works, aa total
total of [zJ
7 integer
integer values
values..
We
absolute value
equation like
can be
We could've
could’ve also
also solved
solved this
this problem
problem algebraically.
algebraically. Any
Any absolute
value equation
like the
the one
one above
above can
be
written as
written
as
< Xx <
< 4
-‐44 <
and since
since x is an
an integer,
integer,
and
- 3 :S:X '.S3
EXAMPLE.2:
2: How
H o w many
many integer
integer values
values of x satisfy
satisfy Ix
ix +
+ 11!I < 5?
5?
EXAMPLE
Here
So
Here we
we go through
through the
the same
same process.
process. The
The largest
largest possible
possible integer
integer for xx is 3 and
and the
the smallest
smallest is -‐ 55.. So
S xx :S:
S 3,
3, aa total
total of []9 possibilities.
possibilities.
-‐55 :S:
Solving algebraically,
algebraically,
Solving
+ l1 <
-‐55 < Xx +
<5
Subtracting 1,
1,
Subtracting
< 4
-‐66 < Xx <
§ Xx :S:
§ 3
-7 55 :S:
EXAMPLE 3:
3: For
For which
which of
of the
the following
following values
values of xis
x is l2x
|2x -‐ SI
5| < 0?
O?
EXAMPLE
A) 0
A)O
B) 2.5
B)2.5
C) 5
C)5
There is no
no such
such value
value of x.
D) There
question.. The
The absolute
absolute value
value of something
something can
can never
n e v e r be
be negative.
negative. There
There is no
no solution,
solution, answer
answer I (D)
Trick question
(D) I·.
174
THE COLLEGE
THE
COLLEGE PANDA
PANDA
EXAMPLE 4: A manufacturer
manufacturer of cookies
cookies tests
tests the
the weight
packages to
to ensure
ensure consistency
EXAMPLE
weight of its
its cookie
cookie packages
consistency
the product.
product. An acceptable
package of cookies
cookies must
must weigh
weigh between
18 ounces
in the
acceptable package
between 16
16 ounces
ounces and
and 18
ounces as
as it
is the
the weight
cookie package,
which of the
comes
out of production.
comes out
production. If w
tois
weight of an
an acceptable
acceptable cookie
package, then
then which
the following
following
inequalities
?
inequalities correctly
correctly expresses
expresses all
all possible
possible values
values of w
w?
A ) Jw-171
[ w ‐ 1 7 |>> 1
A)
1
B )Jw| w ‐ 116J
6 |<< 22
B)
C )Jw+
| w +17J
1 7>
[ >1l
C)
D )Jw-1
| w ‐ l 71
7 |<< 11
D)
In these
types of
of absolute
absolute value
value word
word problems,
problems, start
start with
of the
the desired
17in
this case,
1n
these types
with the
the midpoint
midpoint of
desired interval,
interval , 17
in this
case,
and subtract
subtract it from
from w: Jw
|w -‐ 171.
17|. Think
Think of this
this as
as the
the ”distance,”
away from
midpoint of the
"distance," or ”error,”
"error," away
from the
the midpoint
the
and
interval. We
We don
don’t
this "error"
”error" to
to be
be greater
greater than
than 1 since
then be
be outside
outside the
interval. So
So
would then
the desired
desired interval.
interval.
't want
want this
since w would
ouranswer
our
answer is §}
|w‐
17| <
lw
- 171
1.
1.
We
Remember that
that the
end result
result should
should be
w < 18.
We can
can confirm
confirm this
this answer
answer by
by solving
solving the
the inequality.
inequality. Remember
the end
be 16
16 < w
Let's
Let’s see
see if
if our
o u r answer
answer gives
gives us
us that
that result
result when
when we
we isolate
isolate w.
|w‐l7|<1
lw
- 111 < 1
-‐ 1l < w
w ~- 1177 << 1
Adding
Adding 17,
16 << w
w < 18
16
We
We have
have confirmed
confirmed that
that ~
is the
the correct
correct answer.
answer.
This is the
the graph
graph of y = x:
This
y
N o w this
this is the
graph of y =
Now
the graph
|x|:
!xi:
y
See how
the graph
an y function
function makes
makes all the
the negative
how the
graph changed?
changed? Taking
Taking the
the absolute
absolute value
value of any
negative y-values
y‐values become
become
reflected across
across the
the x-axis).
x-axis). All the
the positive
y-values
positive
positive y-values
y‐values (points
(points in the
the quadrants
quadrants III
I I I and
and IV are
are reflected
positive y-values
stay
they are.
that you
you should
able to recognize.
recognize .
stay where
where they
are. This
This V-shape
V-shape is the
the classic
classic absolute
absolute value
value graph
graph that
should be
be able
175
CHAPTER
CHAPTER 20 ABSOLUTE
ABSOLUTE VALUE
VALUE
table of values
transformation . If
then compare
compare fJ (x)
A table
values is another
another way
way to see this
this absolute
absolute value
value transformation.
If fJ (x) = 2x, then
with
|
f
(
x
)
|
with IJ (x)J.
-3
-6
6
X
J (x)
IJ (x)I
- 2
- 1
0
1
2
-4
- 2
0
2
4
4
2
0
2
4
3
6
6
The negative
the positive
positive values
positive .
negative values
values of Jf (x) become
become positive
positive and
and the
values of Jf (x) stay
stay positive.
EXAMPLE
of y = j2x
1j ?
EXAMPLE 5: Which
Which of the following
following could be
be the graph
graph ofy
IZX ‐- 1|?
B)
B)
A)
A)
C)
C)
yy
yy
yy
D)
D)
y31
x
The
absolute value
the absolute
absolute value
value of something
something can
can never
The entire
entire function
function is enclosed
enclosed in an
an absolute
value and
and since
since the
never be
be
negative,
than or equal
words, the graph
must
on or above
above the
negative, y must
must always
always be
be greater
greater than
equal to 0.
0. In other
other words,
graph m
u s t lie on
the x-axis.
That
eliminates (A)
without the absolute
absolute value.
the answer,
answer, we
That eliminates
(A) and
and (8).
(B). In fact, (A)
(A) is the
the graph
graph of 2x -‐ 1 without
value. To get
get the
we
the x-axis so
take all the
take
the points
points with
with negative
negative y-values
y-values in the
the graph
graph of (A)
(A) and
and reflect them
them across
across the
so that
that they're
they’re
[19].
positive. The graph
graph we
we end
end up
up with
with is (C) .
positive.
One great
mentioning here
obtaining points
points that
that
One
great tactic that's
that’s worth
worth mentioning
here is narrowing
narrowing down
down the
the answer
answer choices
choices by obtaining
2(0) -‐ 11|I =
point (0,1)
(0, 1) must
then be
be on
are
are easy
easy to calculate.
calculate. For
For example,
example, ifif we let
let xx =z 0, then
then y =: 1|2(O)
= 1. The
The point
must then
the graph,
that (0.5,
0 ) must
must also
on the
graph . This
the
graph, eliminating
eliminating (A)
(A) and
and (B). Letting
Letting y =
z 0,
0, we
we now
n o w find that
(0.5,0)
also be
be on
the graph.
This
eliminates
should only
only have
eliminates (D)
(D) because
because (D)
(D) has
has two
t w o x-intercepts
x‐intercepts wherea
whereass the
the graph
graph should
have one
one..
176
THE COLLEGE
THE
PANDA
COLLEGE PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 320.
A calculator should
should N
O T be used
NOT
used on the
following
following questions.
questions.
yy
llff fJ(x)
(x) =
= -‐2x2
2x 2 -‐ 3
3xx+
+ 11, ,what
what iissthe
the value
value o
off
1
/(1)
1?
|f(1)|?
x
A) 3
B) 4
C) 5
D) 6
Which
Which of
of the
the following
following could
could be
be the
the equation
equation of
of
the
xy-plane above?
the function
function graphed
graphed in
in the
the xy-plane
above?
A) ) yy =
x l -‐ 2
A
= -l‐ M
B)
y
=
lx
2
B ) y = hl h- 4
q
y
=
lx
l
+
C ) y = h H 42
D) ) yy=: Ix
D
M - ‐ 21
fl
If
x is aa positive
what is
is
If 1|2
2 -‐ x]
x i > 5 and
and xis
positive integer,
integer, what
the minimum
minimum possible
the
possible value
value of
of x
x?
?
If |x
10, which
Ix ‐- 3]
3 1 > 10,
which of
of the
the following
following could
could be
be
the
the value
value of |xl
lx l ?
?
A
A) ) 2
m4
B) 4
Which of the
Which
is equal
the following
following expressions
expressions is
equa l to
to
‐5
- 5 for
for some
some value
value of x ??
A
A))
B
B))
C
q)
D
D))
C) ) 6
C
D
D)) 8
| x-‐ 661
| + 22
Ix
| x- ‐ 221-| ‐ 6
Ix
6
| x++ 2
+ 66
Ix
21| +
A calculator is allowed on the following
questions .
[ x++61
6 |-‐ 2
Ix
2
H
o w many
different integer
integer values
How
many different
values of
of x satisfy
satisfy
|x+
6I<
3?
Ix + 61< 3?
177
VALUE
ABSOLUTE VALUE
CHAPTER 20 ABSOLUTE
CHAPTER
If In - 21 = 10, what is the sum of the two
possible values of n ?
y
A) 4
B) 6
C) 12
D) 20
the
the function
graph of the
function f is shown
shown in the
The graph
be
following could
the following
Which of the
above. Which
xy-plane above.
xy-plane
could be
?
I
IJ(x) ?
function y := |f(x)|
the
the function
graph of the
the graph
A)
A)
which of the
then which
where x < 10, then
101=
If |x
= b, where
Ix ‐- 10|
equivalent to bb -‐ x ??
following is equivalent
following
y
A)) ‐- 110
A
0
10
B) 10
2b - 10
C) 217‐10
C)
100-‐ 22b
D)) 1
D
b
B)
y
hot dogs
that its hot
ensure that
must ensure
dog factory must
hot dog
A hot
dogs
length .
inches in length.
6~ inches
and 62
inches and
6~ inches
between 6};
are
are between
this factory,
hot dog
length of a hot
the length
lfIf h is the
dog from this
inequalities
following inequalities
the following
which of the
then which
then
?
h7
values of h
accepted values
expresses the accepted
correctly expresses
correctly
C)
y
A)
lh- 6~I< ~
B)
\h- 6~\< 1
C)
\h- 61\< i
D)
11,- 611> i
D)
y
178
THE COLLEGE
COLLEGE PANDA
THE
PANDA
In - 21< 5
bakery standardizes
A bakery
standardizes muffins
muffins to weigh
weigh
1
between 11Z
between
[f m
weight of
~3 and
and 21
2 ! ounces.
ounces. If
m is the
the weight
How many
many integers
How
inequality
integers n satisfy
satisfy the
the inequality
above?
above?
4
4
a muffin
muffin from
a
this bakery,
following
from this
bakery, which
which of the
the following
inequalities
m ??
inequalities expresses
expresses the
the possible
possible values
values of m
A) Six
B) Seven
Seven
3
1
A)
A) m‐11‘<Z
lm-1~
1<~
Eight
C) Eight
D)) N
Nine
D
ine
1
B) Im- 21<B)|m‐2|<1
4
1
C)|m‐2|<1
C) Im- 21<22
tape must
Rolls of tape
certain length.
length.
must be
be made
made to a certain
They must
must contain
contain enough
tape to cover
cover
enough tape
between 400 feet and
between
and 410 feet. If
If/I is the
the length
length of
a roll
roll of tape
tape that
a
that meets
meets this requirement,
requirement, which
which
of the
the following
following inequalities
inequalities expresses
expresses the
the
possible values
?
possible
values of I ?
D)
|I -‐ 400|
10
A) I/
4001< 10
B) I/
|I -‐ 405|
B)
4051> 5
C) II
U+
+ 405|
4051 < 5
D) H
I/ -‐ 4051
4051<
<5
HIlllllllllllllilllllllll
If l4x
|4x -‐ 4|
|5y +
+10]
= 15, what
what is
41= 8 and
and l5y
101=
is the
smallest possible
possible value
smallest
value of xy ??
A)
A ) ‐- 220
0
B)) -‐ 115
B
5
C)) -‐ 5
C
D
D)) -‐ 1
If lal
la} < 1,
1,then
which of the
then which
the following
following must
must be
be
true?
true?
1
I. - > 1
L1>1
a
2
ll
. a¥
1
IL
<1
III .Laa>> -‐ 1l
H
A)
B
B))
C)
D))
D
[ I ] only
III
only
n d 11
11only
1l aand
only
II and
I I only
and III1
II,, JI,
I I , and
and IIII
II
179
3
1
lm-}~1<1
21
Ang
Angles
les
Exterior Angle
Angle Theorem
Theorem
An exterior
is extended.
the triangle
triangle below,
below, xx°0 designates
designates an
an
exterior angle
angle is
is formed
formed when
when any
any side
side of
of a
a triangle
triangle is
extended . In
ln the
exterior
angle.
exterior angle.
s u m of
the two
t w o angles
angles in
furthest from
In this
An exterior
exterior angle
angle is always
always equal
equal to the
the sum
of the
in the
the triangle
triangle furthest
from it.
it. In
this case,
case,
xI z= aa ++ b
EXAMPLE 1:
D
A
100°
C
3x°0
3x
E
E
B
What is the
value of
the figure
figure above?
What
the value
of x in the
above?
A
C E must
be 80°.
o w there
are aa lot
lot of
of ways
ways to
to do
do this,
is the
LD
DC£
must be
80°. N
Now
there are
this, but
but using
using the
the exterior
exterior angle
angle theorem
theorem is
the
fastest:
fastest:
80 ++ Xx =: 3x
80
3X
80 =: 2x
80
r=E
180
THE
PANDA
THE COLLEGE
COLLEGE PANDA
Parallel Lines
Parallel
711
When
the following
following are
are true:
true:
When two
two lines
lines are parallel,
parallel, the
0
angles are
(e.g. £1
= 44)
• Vertical angles
are equal
equal (e.g.
L'.l =
L 4)
0
: £5
= 46)
• Alternate
Alternate interior
interior angles
angles are equal
equal (e.g.
(e.g. 44
L4 =
L S and
and 43
L3 =
L 6)
0
41l =
: L
45)
• Corresponding
Corresponding angles
angles are
are equal
equal (e.g.
(e.g. L
S)
0
supplementary (e.g.
180°)
• Same
Same side
side interior
interior angles
angles are
are supplementary
(e.g. 43
L3 +
+ 45
LS := 180°)
No need
memorize these
just need
need to know
w o parallel
need to memorize
these terms.
terms . You just
know that
that when
when ttwo
parallel lines
lines are
are cut
cut by
by another
another line,
line,
there are
are two
there
t w o sets
sets of equal
equal angles:
angles:
L 1 = L'.'.
4 = LS = LB
L2 = L'.'.
3 = L6
= L7
EXAMPLE 2:
EXAMPLE2:
C
A
G
F
E
In
the figure above‐If
above, AC ||
Inthefigure
ACE=
=40°,whatisthevalueofx?
II"CfiandE‐F'
GD and BF HE.
II CE. IfACAE
If LCAE =
= 70°
70° and
and [LACE
400, what is the value of x?
Here
fastest way: L
Here is the fastest
[ A
C E = .LABF
AABF =
: 40° because
because they are corresponding
lines
ACE
corresponding angles
angles (A‐C
(AC cuts
cuts parallel
parallel lines
BF and
and fiCE).
W
) . Since angle
an exterior
exterior angle
angle to 6A AABF
B F,, x =
z 70 +
+ 40 =
= I‑ 110° 1angle x is an
181
CHAPTER
ANGLES
CHAPTER 21 ANGLES
Polyg ons
Polygons
Triangle
Triangle
180°
180°
Quadrilateral
Quadrila tera 1
Pentagon
Pentagon
00
□
360°
540°
Hexagon
Hexagon
720°
angles by 180°
interior angles
sum of the interior
increases the sum
As you
additional side
side increases
180°..
each additional
above, each
polygons above,
the polygons
from the
can see from
you can
angles is
interior angles
the interior
For any
u m of the
polygon, the ssum
any polygon,
sides
number of sides
180(n -‐ 2)
2) where
where n is the number
lBO(n
1080° .
= 180 x 6 := 1080°.
180(8 ‐- 2) =
angles is 180(8
interior angles
sum of the
the sum
sides, the
has 8 sides,
50
the interior
which has
octagon, which
an octagon,
So for an
regular . If
are regular.
above are
shown above
polygons shown
equal. The polygons
are equal.
sides and
which all sides
one in which
A regular
and angles
angles are
polygon is one
regular polygon
.
135°
=
-;-8
1080°
of
measure
a
have
would
angle
interior
each
regular,
were
octagon
our
our octagon were regular, each interior angle would have a measure
+ 8 : 135°.
drawing
triangles by drawing
several triangles
into several
up into
split up
be split
polygon can be
that any
from the fact that
formula comes
lBO(n ‐- 2) formula
The 180(n
comes from
any polygon
others .
the others.
vertex to the
any oone
from any
lines from
n e vertex
Count for
sides. Count
number of sides.
than the number
results from
that results
triangles that
number of triangles
The number
from this process
process is always
always ttwo
w o less than
- 2),
(n ‐~
180°
be
must
polygon
a
within
angles
the
contains 180°,
triangle contains
each triangle
Because each
yourself! Because
yourself!
180°, the
the sum
s u m of
angles within a polygon m u s t be 180°(n
sides.
where
number of sides.
where n is the number
EXAMPLE3:
EXAMPLE3;
?
value of x '?
above. What is the value
figure above.
extended as
pentagon are extended
Two sides
as shown
shown in the figure
sides of a regular pentagon
be 540° -;-5
angle must
interior angle
each interior
So each
540° . So
5 -‐ 2) =
pentagon is 180(
a pentagon
degrees in a
number of degrees
total number
The total
180(5
2 540°.
m u s t be
+ 5 = 108°
108°..
72°.
=
108
180
be
must be
formed by the
triangle formed
the triangle
within the
The angles
the intersecting
intersecting Jines
lines must
‐ 108 = 72°.
angles within
108°
108°
So,x=180‐72‐72:-.
So, x = IBO - 72 - 72 = j 36° 1182
THE COLLEGE PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 322.
should N
NOT
A calculator should
O T be
be used
used on the
following questions.
questions.
following
c.....----~-----------1
kO
Note: Figure
Figure n
not
Note:
o t drawn
scale..
drawn to scale
In
m are
parallel.
1n the
the figure
figure above,
above, lines
lines I and
and m
are parallel.
Whatisthevalueofa+b+c+d?
What is the value of a + b + c + d ?
1n the
the figure
figure above,
In
above, 1”
= 50
50 and
= 140. What
i=
and k =
What is
the value
the
value of j?
j ?
A)) 660
A
0
A) 270
B)
B) 70
70
B) 360
C) 80
80
C)
C) 720
D)) 9900
D
cannot be
D) It cannot
be determined
determined from
from the
the
information given.
information
given.
Note: Figure not drawn to scale.
Note
Figure not
Note:: Figure
scale..
not drawn
drawn to scale
1n the
In
= 40, what
the figure
figure above,
above , ifif x =
what is the
the value
value
ln the
the figure
In
figure above
above,, what
what is the
the value
value of y ??
of yy??
A)) 330
A
0
A ) 440
0
A)
B)) 4400
B
B) 50
B)
C)) 550
C
0
C) 80
C)
80
D
D)) 7700
D)) 9900
D
183
183
CHAPTER 21
CHAPTER
21 ANGLES
ANGLES
A calculator is allowed
following
allowed on the following
questions.
questions.
In the figure above, lines I and m are parallel.
What is x in terms of a and b ?
Note: Figure
Figure n
not
drawn to scale.
scale.
Note:
o t drawn
A) a +
+b
In the
the figure
figure above,
above , what
what is the
the value
va lue of x +
In
+ yy??
A
125
A)) 125
B)
B) a -‐ b
C
C)) b -‐ a
D
1 8 0- ‐aa-‐ b
D)) 180
B
B)) 180
180
C) 235
D) 280
r»
280
_
70°
11°
30°
17°
Note: Figure
drawn to
Note:
Figure not
not drawn
to scale
scale..
In the
figure above,
what is the
the figure
above, what
the value
value of a +
+ b ??
A ) 880
0
A)
B) 100
C) 110
C)
D) 120
Note : Figure
Figure nnot
m
o t drawn
drawn to scale
scale..
In the
the figure
figure above,
ln
above, what
what is the
the value
value of zz ??
A
5
A)) 335
B) 45
45
C)
C) 55
55
D
0
D)) 880
184
THE COLLEGE
THE
COLLEGE PANDA
PANDA
(x + 40)° x°
the figure
In the
figure above,
above, what
what is
is the
the value
value of
of x
x?
?
A) 60
60
A)
70
B) 70
C) 75
75
C)
In
In the
the figure
figure above,
above, a
a rectangle
rectangle and
and a
a
quadrilateral overlap.
quadrilateral
u m of
overlap . What
What is
is the
the ssum
of the
the
degree measures
degree
measures of the
the shaded
shaded angles?
ang les?
D)
D) 80
80
I_
A)
A) 360
360
B) 540
C) 720
D) 900
I -'1
Ar
',
~•
4
~
"
:.·•.
~
•
•
\
.,
regular hexagon
hexagon is shown
A regular
figure above
above..
shown in
in the
the figure
What is the
What
the value
value of x ?
?
A) 15
A) 15
Note: Figure
Figure n
m
o t drawn
drawn to scale
scale..
not
B) 20
20
C)
25
C) 25
the figure
figure above,
above, what
In the
value of y?
y?
what is the
the value
A)
A) 100
B) 130
D) 30
C) 140
D) 150
185
"'v,)
CHAPTER 21 ANGLES
Note: Figure
Figure not
not drawn
drawn to scale.
Note:
scale .
the figure
figure above,
above, lines
lines I and
and m
In the
m are
are parallel.
parallel.
Which of the
the following
following m
Which
u s t be
must
be true?
true?
Note: Figure not drawn to scale.
I,
= 3b
3b
I. a =
In the
the figure
figure above,
above , lines
lines I, m,
m, and
and n are
are parallel.
parallel.
II . a +
c
11.
+ b=
= b+
+ C
What is the
the value
What
value of a +
+ b ??
45
III. bb == 45
A)
A) IIllonly
I I only
8) Il and
and [I
II only
only
B)
C) II
ll and
and 111
lil only
only
C)
D)) I,
D
1 ,[II,
L aand
n d III
111
Note: Figure not drawn to scale .
the figure
figure above,
above, what
what is the
In the
value of xx +
+ yy??
the value
A
0
A)) 110
B ) 220
0
8)
C
0
C)) 330
D
0
D)) 550
186
22
Triangles
Triangles
& Equilateral Triangles
Isosceles
Isosceles 8.EquilateralTriangles
equal.
those sides
opposite those
The angles
length. The
equal length.
An isosceles
isosceles triangle
has ttwo
w o sides
sides of equal
angles opposite
sides are
are equal.
that has
one that
triangle is one
B
Because AB =
: AC,
AC, L.C
4C =
: AB.
L.B.
Because
angles
angles, the
imply equal
sides imply
equal sides
Because equal
length. Because
same length.
the same
triangle, all sides
equilateral triangle,
an equilateral
In an
sides have
have the
equal angles,
the angles
60°.
are
are all 60“.
degree measure
the degree
What is the
measure of 50°. What
has a measure
one of the
isosceles triangle,
an isosceles
EXAMPLE 1: In an
triangle, one
the angles
angles has
measure
triangle?
the triangle?
the greatest
possible angle
angle in the
greatest possible
of the
are ttwo
There are
angles. There
equal angles.
also ttwo
triangle has
isosceles triangle
An isosceles
has nnot
o t only
only ttwo
w o equal
equal sides
sides but
but also
w o equal
w o possibilities
possibilities for
the
triangle, or the
a 50-50--80
making a
could be
angle could
an
i t h an
an angle
angle of 50°.
503. Another
Another angle
be 50°,
50C, making
50‐50‐80 triangle,
with
triangle w
isosceles triangle
an isosceles
80° Iis the
possibilities, j 800
these two
Given these
triangle . Given
other
w o angles
making aa 50-6.5-<>5
50‐65‐65 triangle.
t w o possibilities,
the greatest
greatest
equal, making
be equal,
could be
angles could
other ttwo
possible
triangle.
the triangle.
angle in the
possible angle
187
TRIANGLES
CHAPTER 22 TRIANGLES
CHAPTER
w a s
C
A
B
B
,
is the value of j + k + l + m + n + o ?
· lS eq ·.. teral. What
maxe‘i‘fggm‘meghemieucmequflatem.
Whatisthevalueofj+k+l+m+n+o?
,ih1~}
i;:~ I
degree
total degree
triangles has
these triangles
Each of
one. Each
the equilateral
triangles within
Solution
are 3 smaller
smaller triangles
within the
equilateral one.
of these
has aa total
There are
Soluti on 1: There
we
what
get
to
ACB
L
out
total of 180° x 33 = 540°. We
combined total
measure
We need
need to subtract
subtract out L A CBto get what we want.
want.
180°, for aacombined
measure of 180°,
is 60°. So
ACB
equilateral, L
Because
triangle ABC
ABC is equilateral,
[ A
C Bis
80 540° -‐ 60°
60° = I-.480°
Because triangle
I.
line, they
straight line,
a straight
form a
and II form
k and
Because k
both j and
is equilateral,
on 2: Because
Solution
Because .0-A
AA BC
BCis
equilateral, both
and o0 are
are 60°. Because
they add
add
Soluti
all oour
180°. Adding
up to
add up
also add
and n also
m and
up to 180°. Because
up
Because m
also form
form aa straight
straight line,
line, they
they also
to 180°.
Adding up
up all
u r values,
values, we
we
get
: I-.480° 1get 60° + 180° + 180° + 60° =
Triangles
Right Triangles
angle).
right angle),
the right
opposite the
(the side
made up
triangles are
Right
are made
up of two
t w o legs
legs and
and the
the hypotenuse
hypotenuse (the
side opposite
Right triangles
C
a
b
the legs
of the
lengths of
the lengths
bare
and b
a and
where a
c2 , where
b2 = C2,
theorem: a£122 + b2=
obeys the
Every
triangle obeys
the pythagorean
pythagorean theorem:
are the
legs and
and
right triangle
Every right
hypotenuse.
c is the
the hypotenuse.
length of the
the length
188
THE COLLEGE
THE
PANDA
COLLEGE PANDA
EXAMPLE
EXAMPLE 3:
The
rectangle above
has a
length 20.
is twice
twice as
as long
long as
as the
height,
The rectangle
above has
a diagonal
diagonal of
of length
20. If
If the
the base
ba se of
of the
the rectangle
rectangle is
the height,
what
the height?
height?
what is the
The
w o right
be xx and
be 2x.
2x. Using
Using the
The diagonal
diagonal of
of any
any rectangle
rectangle forms
forms ttwo
right triangles.
triangles . Let
Let the
the height
height be
and the
the base
base be
the
pythagorean theorem,
pythagorean
theorem ,
x2+
(2x)22 = 202
x2 + (2x)
202
2
xx2+4x2=400
+ 4x 2 = 400
2
5x2=400
5x
= 400
2
xx2=80
= 80
M
If you
take the
the SAT enough
If
you take
enough times,
what you’ll
find is that
repeatedly. For
times, what
you'll find
that certain
certain right
right triangles
triangles come
come up
up repeatedly.
For
example,
the 3-4-5
3‐4‐5 triangle:
example, the
triangle:
4
A set of three
three whole
whole numbers
numbers that
triple.. Though
Though not
not
that satisfy
satisfy the
the pythagorean
pythagorean theorem
theore m is
is called
called a
a pythagorean
pythagorean triple
necessary, it’ll
it' ll save
save you
you quite
necessary,
and improve
improve your
you learn
recognize
the
common
quite a
a bit
bit of time
time and
your accuracy
accuracy ifif you
learn to
to recognize the common
triples
that show
show up:
triple s that
up:
3,
4, 5
3,4,5
6,
8, 10
6,8,10
5,12,
13
5,
12,13
7, 24, 25
7,24,25
8,
15, 17
8,15,
Note that
that the
the 6‐8‐10
Note
just aa multiple
of the
6-8-10 triangle
triangle is just
multiple of
the 3‐4‐5
3-4-5 triangle.
triangle .
189
CHAPTER 22 TRIANGLES
CHAPTER
TRIANGLES
Special Right
Special
Right Triangles
Triangles
will have
have to memorize
You will
w o special
right triangle
45°‐45°‐90°:
memorize ttwo
special right
triangle relationships.
relationships. The first is
is the 45°-45
°-90 ° :
45°
X
45°
X
The best
that it's
it's isosceles-the
isosceles‐the ttwo
w o legs
best way
way to think
think about
about this
this triangle
triangle is that
legs are
are equal.
equal. We
We let
let their
their lengths
lengths be
be x.
x.
The hypotenuse,
which is always
always the biggest
biggest side
side in
in aa right
right triangle,
u t to
hypotenuse, which
triangle, turns
turns oout
to be
be fl
/2 times
times x.
x.
We can
can prove
We
relationship using
theorem, where
prove this relationship
using the
the pythagorean
pythagorean theorem,
where h
11is
is the
the hypotenuse.
hypotenuse .
x2
+ x2 =
112
x2+x2
zh2
2
2x2=h2
2x
= 112
x/27:\/h‘2
v'2li = ~
xx/Ezh
xV2= 11
l show
show you
proofs not
1
you these
these proofs
n o t because
because they will be
be tested
tested on
on the
problem-solving
the SAT,
SAT,but
but because
because they
they illustrate
illustrate problem-solving
concepts that
that you
have to use
use on
on certain
certain SAT questions.
questions.
concepts
you may
may have
The second
second is the
30°‐60°‐90°:
the 30°
-60 ° -90 °:
60°
X
Xfi
Because 30° is the
Because
smallest angle,
from it is the shortest.
side be
be x.
x. The hypotenuse,
hypotenuse,
the smallest
angle, the side
side opposite
opposite from
shortest. Let that
that side
the
turns out
to be
be ,/3
\/§ times
times x.
x.
the largest
largest side,
side, turns
turns out
out to be
be twice x, and
and the side
side opposite
opposite 60°
60° turns
out to
One
students make
make is to think
think that
that because
600 must
One common
common mistake
mistake students
because 60° is
is twice
twice 30°,
30°, the
the side
side opposite
opposite 60°
must be
be
twice
as the
the side
side opposite
opposite 30°. That
That relationship
relationship is
is N
O
T
true.
You
cannot
extrapolate
the
ratio
twice as
as big
big as
NOT true . You cannot extrapolate the ratio of
of the
the
sides
the angles.
angles. Yes,
side opposite
opposite 60°
30°, but
but it
it isn't
isn’t
sides from the
the ratio
ratio of the
Yes, the
the side
60° is
is bigger
bigger than
than the side
side opposite
opposite 30°,
twice as
twice
as long.
long.
190
THE COLLEGE
COLLEGE PANDA
PANDA
We can
prove the
relationship by using
equilateral triangle.
triangle . Let
could use
We
can prove
the 30-60-90
30‐60‐90 relationship
using an
an equilateral
Let each
each side
side be
be 2x
2x (we
(we could
use xx
but
why 2x
makes things
but you'll
you’ll see
see why
2x makes
things easier
easier in aa bit)
bit)::
B
A
2x
2x
C
Drawing aa line
down the
middle from
from B
B to A_C
creates ttwo
w o 30--60-90
30‐60‐90 triangles.
Because an
triangle
Drawing
line down
the middle
AC creates
triangles. Because
an equilateral
equilateral triangle
2x was
was used
avoids any
fractions.
is symme
trical , AD
symmetrical,
AD is half
half of 2x, or just
just xx.. That's
That’s why
why 2x
u s e d- ‐ it
i t avoids
any fractions.
B
D
X
A
To find BO,
pythagorean theorem:
BD, we
we use
use the
the pythagorean
theorem:
AD22 + B0
3022 = A82
AD
AB 2
2
BD22
xx2
+ 80
= (2x)2
(2x)2
31322 =
: (2x)2 -‐ x2
80
x2
3022
80
2
: 4x2
=
4x 2 -‐ xx2
3022
80
= 3x2
3x 2
M 2: = ~
@
✓ao
BD =
= X\/§
BO
xV3
Triangle
the 30--6090 relation
ship .
Triangle ABO
ABD is proof
proof of the
30‐60‐90
relationship.
1
9]
191
C
CHAPTER 22 TRIANGLES
CHAPTER
TRIANGLES
Wu 4:
A
B
C
WintistheareaofAACBshownabwe?
the area of ..6.ACBshown above?
Wbatis
A)-y2.
AN?
'8)2-/2
ism/i
C) 4
04
D) 8 ·
D18
leg).
each leg).
than each
greater than
times greater
v'2Z times
is \/‐
hypotenuse 1s
(the hypotenuse
triangle relationship,
45--45-90 triangle
the 45‐45‐90
Using the
USing
relationShiPr AC
AC =
= BC
BC‐= 7~ (the
‐ )(x/i‐ ) 22 (_>
22
22 (1/5
1 4
4 ) ( 4
1 16
4 ) =
. then
The area
z _1 ( 16) =
z 4.
v'2
then ‐1 ( v'2
1s
area is
Answer
Answer ~(C) .
EXAMPLE5:
EXAMPLES:
A
10
I
D
=
C
B
=
ratio of AC to CB ?
the ratio
What is the
AB =
30° apd A8
LB = 30°,and
above, AD
the £igur-e
hi,
In the
figure above,
AD = DC,
DC, 48
= 10
10... What
V2
1/5
«5
2
“W7 ”7th 0m ”5
triangle
30-60-90 triangle
is a 30‐60‐90
6 ADB
triangle . A
45-45-90 triangle.
only isosceles
not only
is not
6 ADC
DC, A
Because
A D C15
isosceles but
but also
also a 45‐45‐90
A D BISa
ADD ‐=‐ DC,
Because A
Using
5v'3.
DBB‐=‐ 5\/§.
and D
hypotenuse, 5, and
half the hypotenuse,
is half
AD 15
relationship, AD
30-60-90 relationship,
the 30‐60‐90
Using the
with
3Using
hypotenuse of 10. Using
a hypotenuse
with a
DCC=
DBB- ‐ D
CBB‐=‐ D
and C
5, and
=‐ 5,
DCc _
5v'2, D
ACC=
relationship, A
the
‐‐ 5\/§,
‐ ‐ 5y'3
5f3‐- 5.
5
45--45-90 relationship,
the 45-45‐90
AC
A_c_
=
5y'2
5\/§
CBB 5v'3
C
s fi ‐- s5
Answer
Answer ~(A) .
192
_
=
y'2
W2
fiv'J-‐ 1i
THE COLLEGE
PANDA
THE
COLLEGE PANDA
Similar
Similar Triangles
Triangles
When ttwo
triangles have
have the
the same
their sides
sides are
When
w o triangles
same angle
angle measures
measures,, their
are proportional:
proportional :
B
A
C
the figure
figure above,
above, 6
DBE and
6 AABC
In the
ADBE
and A
B C both
both have
have right
parallel to A‐C,
right angles
angles and
and share
share AB.
L.B. Because
Because W
DE is parallel
AC,
[L.BED
B E D is equal
equal in measure
measure to ABCA.
Therefore, 6
ADBE
B C have
sets of angles
angles and
and are
are
L.BCA. Therefore,
DBE and
and A
6 AABC
have congruent
congruent sets
similar to each
each other.
other words,
just a
AABC.
has the
the same
same shape
shape but
but not
not
similar
other. In other
word s, 6ADBE
DBE is just
a smaller
smalJer version
version of 6
ABC. It has
the same
same size.
size. If
we draw
draw the
the ttwo
triangles separately
the
If we
w o triangles
separately and
the sides
sides some
some arbitrary
arbitrary lengths,
lengths, we
we can
can see
see
and give
give the
this more
this
more clearly.
clearl y.
B
B
4
8
D
A
5
10
6
6
3
E
C
Notice that
”ma‐tches
up”" with
with E,
with E.
math
Notice
that m
AB "ma
tches up
BD, AE_matches
AC matches up with
with W,
DE, and
and Ematches
BC matches up
up with
BE. Using
Using math
terms,
say that
corresponds
BD, AC correspo
corresponds
and
BC
corresponds
with
BE.
term
s, we say
that AB correspo
nds with BO,
nds with DE,
DE, and
corresponds
BE.
For illustrative
illustrative purposes,
we made
the sides
sides of the
the big
as long
the sides
sides of the
the smaller
smaller one.
one. But
purposes, we
made the
big triangle
triangle twice
twice as
long as
as the
regardless
what the
thing to
remember is
is that
that the
the cor
corresponding
sides of
of
regardless of what
the actual
actual numbers
numbers are,
are, the
the important
important thing
to remember
responding sides
similar
ratios of
of their
their lengths
lengths are
are equivalent.
In oour
u r example,
example,
similar triangles
triangle s are
are proportional,
proportional, so
so the
the ratios
equivalent. In
Q _ E _ E _
BD_DE_BE_
Forming
these types
types of
of equations
is your
your goal
goal in
in every
triangles.
Forming these
equations is
every SAT question
question dealing
dealing with
with similar
similar triangles.
193
CHAPTER 22 TRIANGLES
CHAPTER
TRIANGLES
EXAMPLE6:
EXAMPLEG:
B
A
C
drawn to scale.
Note: Figure
Figure not
not drawn
Note:
DB =
AD = 99,, DB
lnAABCabove,fiE‐isparalleltoza
= 33,, aand
n d DE
DE= 22..
.ti.ABC above, DE is parallel to AC, AD
In
1: What is the length of AC1
PART
Pmnwmetheimgmofzc?
?
area of l:i.BAC?
the areaofABAC
PART 2: What. the ratiooftheareaofABDE-Ito
ratio of the ai:ea of ABD E to the
rakrziwhamthe
1
1
.1
1
!
B)
A)!
Al‘s.
3
)
;
C);
Dlfi
16
D)
9
C)
•
4
3
similar
see similar
they see
whenever they
make whenever
students make
that students
mistake that
common mistake
6. A common
NOT 6.
answer is NOT
Part 1 Solution:
Solution: The
The answer
many
course,
Of
not.
do
they
when
other
each
certain segments
that certain
assume that
triangles is to assume
triangles
segments correspond
correspond with
with each other when they do not. Of course, many
assume
to
easy
it's
case,
particular case, it's easy to assume that
this particular
In this
make . In
easy to make.
mistake is easy
this mistake
that this
so that
designed so
are designed
questions are
questions
that
ratios
the ratios
at the
look at
to look
have to
We have
incorrect. We
be incorrect.
would be
times DE,
be 3 times
must be
BO, AC must
because
because AD is 3 times
times BD,
DE, but
but this
this would
triangle).
any triangle).
of any
side of
a side
not a
is not
AD is
that A‐D
(Note that
portions of those
triangles, not
the triangles,
sidesof the
between the sides
between
not portions
those sides
sides (Note
parallel,
are parallel,
AC are
DE and
Because D‐E
ratios . Because
correct ratios.
up the correct
we set up
and make
beginning and
the beginning
from the
let's start
So let’s
So
start from
make sure
sure we
and KC
the
Equating
similar.
are
.6.BAC
and
.6.BDE
,
Therefore
L.BCA.
to
equal
is
equal to L.BAC
L.BDE
ABDE is equal
ABAC and
and L.BED
[ B E D equal
ABCA. Therefore, ABDE and A B A C are similar. Equating the
corresponding sides,
relevant corresponding
the relevant
ratios
sides,
ratios of the
DE
BD _ DE
BO
E
X _AC
R
BA
2
33 __2
= R
35 +? 9) _ AC
2
3
fi12 == AC
E
Cross
multiplying ,
Cross multiplying,
3AC =
= 24
24
3AC
AC:‑
AC =~
triangles .
with
working w
when working
take when
there are
warning, there
word of warning,
a word
As a
are valid
valid shortcuts
shortcuts that
that some
some students
students take
i t h similar
similar triangles.
DE:AC.
to
equal
not
is
80:0A is not equal to DEzAC. II
that BD:DA
though we
BE:EC, even
equal to BEzEC,
80:0A is equal
example, BDzDA
For example,
For
even though
we just
just showed
showed that
misused . Just
easily misused.
are easily
they are
because they
shortcuts altogether
types of shortcuts
these types
you avoid
that you
recommend that
recommend
avoid these
altogether because
Just focus
focus
triangles
similar triangles
handle any
to handle
able to
be able
you'll be
and you'll
right ratios
up the right
on setting
setting up
ratios between
between corresponding
corresponding sides
sides and
any similar
on
throw at
might throw
question
at you.
you.
question the SAT might
of
ratio of
the ratio
of the
the square
equal to the
areas is equal
their areas
the ratio
are similar,
triangles are
When ttwo
Solution: When
w o triangles
similar, the
ratio of their
square of
Part 2 Solution:
2
2
Answer
:4 =
the areas,
ratio of the
the ratio
Squaring that
their
The ratio
ratio of the
the sides
sides is 1:4. Squaring
that ratio,
ratio, we get the
areas, 1
12:42
: 1:16.
1:16. Answer
sides. The
their sides.
o t drawn
scale.
drawn to scale.
figure is nnot
the figure
that the
Note that
/-.(D) /. Note
194
..-
.··
THE
THE COLLEGE
COLLEGE PANDA
PANDA
In
it was
was easy
the triangle
triangle similarity
similarity and
sides corresponded
corresponded with
with each
each
1n this
this example,
example, it
easy to
to see
see the
and determine
determine which
which sides
other.
similar triangles
and their
sides are
are more
more difficult
difficult to
spot
and
keep
other. In
1n tougher
tougher questions,
questions, similar
triangles and
their corresponding
corresponding sides
to spot and keep
track
o t always
track of. The
The ratios
ratios are
are n
not
always obvious.
obvious .
For these
labeling equivalent
with tick
marks. Sides
Sides opposite
from
For
these tougher
tougher questions,
questions , II recommend
recommend labeling
equivalent angles
angles with
tick marks.
opposite from
angles
with the
the same
same number
number of
of tick
tick marks
marks will
w i l l correspond
correspond with
each other.
an example
angles with
with each
other. Here’s
Here 's an
example to
to illustrate:
illustrate:
EXAMPLE 7:
EXAMPLE
G
Lh
F
K
H
hithefigtmeabove,whatisthelengthofffi?
In the figureabove, what is the length of KH ?
At first glance,
glance, this
n o t look
like aa similar
similar triangles
triangles question.
of the
the outside
outside
At
this does
does not
look like
question. But
But ifif we
we label
label the
the angles
angles of
triangle
FGH, something
something interesting
tick mark
mark and
and L
[HH with
t w o tick
tick
triangle FGH,
interesting happens.
happens. Let’s
Let's first label
label [F
L F with
with one
one tick
with two
marks.
marks.
/n
A
F
G
G
G
m&
‐>
K
K
H
F
K
H
N o w whenever
w o angles
are equal
w o angles
another, the
the third
third angles
angles must
must
Now
whenever ttwo
angles of one
one triangle
triangle are
equal in measure
measure to ttwo
angles of another,
also
be equal.
Because outside
FGH and
on the
the left both
have right
angles and
share
also be
equal. Because
outside triangle
triangle FGH
and triangle
triangle FGK
FGK on
both have
right angles
and share
L F, AFGK
L FGK must
AF,
m u s t have
(two tick marks).
Likewise, outside
F G H and
triangle
have the
the same
same measure
measure as
as LAH
H (two
marks). Likewise,
outside triangle
triangle FGH
and triangle
GKH on the
the right
GKH
right angles
and share
share AH,
so L.KGH
ZKGH must
have the
the same
same measure
measure as
AF (one
(one
right both
both have
have right
angles and
L.H, so
must have
as L.F
tick mark).
mark).
The
result is that
The result
that the
the outside
triangle, the
the triangle
triangle on
on the
the triangle
on the
right all
all have
have the
the same
same
outside triangle,
the left,
left, and
and the
triangle on
the right
angle
angle measures
measures.. They're
They’re all
all similar
similar to one
one another!
another!
Since 6.
AFFGK
GK is aa 3-4-5
3‐4‐5 triangle
with FK
FK =
= 4,
4, we
we can
can n
o w set
similar triangles
triangles
Since
triangle with
now
set up
up an
an equation
equation of
of ratios
ratios using
using similar
AF
GKon
the left
and A
K H on
on the
the right
the length
6. FGK
on the
left and
6.G
GKH
right to find the
length of m.
KH.
KH (opposite
(opposite I-tick
KH
l-tick angle
side triangle)
l-tick angle
angle in left side
side triangle)
angle in right
right side
triangle) __ GK (opposite
(opposite I-tick
triangle)
GK (opposite
(opposite 2-tick angle
GK
angle in right
side
triangle)
FK
(opposite
2-tick
angle
in
left
side
triangle)
right side triangle) (opposite
angle
side triangle)
EA
3
33 ‘4 4
4 ( K H )=: 9
4(KH)
KH
KH =rn
195
CHAPTER
CHAPTER 22
22 TRIANGLES
TRIANGLES
Parallel
Parallel Lines
Lines and Proportionality
Proportionality
Lines
three or
proportional parts.
Lines that
that cut
cut through
through three
or more
more parallel
parallel lines
lines are
are separated
separated into
into proportional
parts.
a
b
n
m
To illustrate,
transversals that
through three
three parallel
parallel lines.
lines. Therefore,
illustrate, lines
lines m
771 and
and n
it above
above are
are transversals
that cut
cut through
Therefore, the
the three
three
parallel
lines divide
parallel lines
divide lines
lines m
m and
and nn proportionally:
proportionally:
a
‘
1 5C
bbd d
This concept
don 't need
need to know
know the
the underlying
underlying proof.
proof .
concept can
can be
be easily proved
proved using
using similar
similar triangles
triangles,, but
but you
you don’t
problems
Just
rule . It's
shown up on past
exams, usually
Just memorize
memorize the
the rule.
It’s not
n o t tested
tested very
very often
often but
but it has
has shown
past exams,
usually in problems
involving
involving aa trapezoid.
trapezoid.
EXAMPLE8:
· EXt\MPLE
8:
A
•~•=-t
~ llt~ ► if;,
'~ ,.
.......
J~
""trr ,..,.
.-,/' :t,n.;'11,•
~~ ~.
-
...
~
l
"
•
>
B
\
I
P
,,.;-..
... ,.
D
I
Q
C
lnflrefigumabave,fi,P§,and‐D'Careparakl.
PointPliesonEandpointhiesonFC.
lfB
Q = 44,,
In
the figure above, AB, PQ, and DC are parallel. Point
P lies on AD and point Q lies on BC. If
BQ
QC=2,andAD=7.5,whatisthelengthofAP?
-QC
= 2, and AD = 7.5, whatis the length of AP?
Because E
and W
cut through
through three
three parallel
parallel lines,
lines, they
they are
are divided
proportionally.
Because
AD and
BC cut
divided proportionally.
£_@-L2
PD_QC_2_
can see, since
since BQ is twice
twice QC, AP is twice
twice PD (a ratio
ratio of 2 to 1). A 2:1 ratio
ratio means
means that
that AP is -2‐3‐1‐
: ~g
- =
As we can
2+1
3
2
the length
length of AD
AD.. AD
AD is then
then ~5 x 7.5 =
: [fil.
I. If you
you prefer
prefer to do
length PD
be x.
the
do things
things algebraically,
algebraically, then
then let length
PD be
‐
2
Then length
length AP
AP is 2x, and
and since
since AP
AP and
and PD
PD sum
sum to AD
A D , 2x
2x + x == 7.5. This equation
z 2.5 and
we get
Then
equation gives
gives x =
and we
AP =
: 2x =
= 2(2.5)
2(2.5) =
: @J
I. .
AP
196
THE
COLLEGE PANDA
THE COLLEGE
PANDA
Radians
radian is simply
simply another
another unit
used to measure
measure angles.
have feet
meters, pounds
and
A radian
unit used
angles. Just
Just as
as we
we have
feet and
and meters,
pounds and
kilograms, we
we have
degrees and
radians.
kilograms,
have degrees
and radians.
7Tradians
radians = 180°
180°
Tr
If you've
you’ve never
never used
used radians
radians before,
before, don
don't
be put
put off
it’s just
just aa number.
number. We
We could’ve
If
' t be
off by
by the
the 7T.
Tr. After
After all,
all, it's
could've
written
written
3.14 radians
radians :::::;
z 180°
180C
instead, but
but everything
typically expressed
expressed in terms
terms of 7r
working with
with radians
radians.. Furthermore,
Furthermore,
Tr when
when we're
we're working
instead,
everything is typically
is only
only an
anapproximation.
given the
conversion factor
factor above,
above, how
radians?
3.14 is
approximation. So, given
the conversion
how would
would we
we convert
convert 45° to
to radians?
7Tradians __ .::
71 d"
450
45° xx Tr radians
180°
4 radians
ra ians
1800
- Z
Notice that
that the
degree units
units (represented
(represented by the
the little
little circles) cancel
u t just
just as
they should
should in any
conversion
Notice
the degree
cancel oout
as they
any conversion
3
problem.. Now
N o w how
how would
would we convert
convert 7”
degrees? Flip
Flip the
the conversion
; to degrees?
problem
conversion factor.
3rr
d.1ans x ---=
-3‐H ra
radians
& 1800
: 2700
0°
2
Tr
radians
7Tradians
You might
be wondering
wondering why
why we
we even
even need
need radians
radians.. Why n
o t just
just stick
this another
another difference
might be
not
stick with
with degrees?
degrees? Is this
difference
between
rest of the
and meters?
meters? Nope
the chapter
chapter on
on
between the
the U.S.
U S . and
and the
the rest
the world,
world, like
like it is with
with feet and
Nope.. As we'll
we’ll see
see in the
circles, some
calculations are
much easier
are expressed
expressed in radians
circles,
some calculations
are much
easier when
when angles
angles are
radians..
EXAMPLE9:
EXAMPLE
9:
y
m
In the
above, line
origin and
and has a
a slope
slope of y/j.
the xy-plane
xy-plane above,
line m
m passes
passes through
through the
the origin
Vi. IfIf point
point A
A lies
lies on
on line
line m
m
and point
point B lies
lies on
on the
x-axis as
as shown,
what is the
the measure,
measure, in radians,
angle AOB?
AOB ?
and
the x-axis
shown, what
radians, of angle
A).::
A);
71:
6
B)?
n
B) n:
5
C).::
D).::
C);
D)?
4
3
7T
7T
197
CHAPTER 22
CHAPTER
TRLANGLES
TRIANGLES
We can
can draw
from A to the
We
draw aa line
line down
down from
the x-axis
x-axis to
to make
make aa right
the slope
slope is v'3,
V3, the
ratio of the
right triangle.
triangle . Because
Because the
the ratio
the
height of this
height
base is always
to 11 (rise
(rise over
over run).
this triangle
triangle to its base
always \/5
v'3to
run).
y
m
This right
right triangle
triangle should
should look
triangle. Angle
the fi,
v'3,so
This
look familiar
familiar to you.
you. lt's
It’s the
the 30-60-90
30‐60‐90 triangle.
Angle AOB
A 0 3 is opposite
opposite the
so its
its
measure is
measure
is 60°. Converting
Converting that
that to radians,
radians,
7T
60 x 180° 2 3
Answer
Answer ~(D) .
198
THE COLLEGE
COLLEGE PANDA
THE
PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 324.
V.
calculator should
should N
NOT
A calculator
O T be
on the
be used
used on
following
following questions.
questions.
X
The lengths
length s of the
the sides
sides of a
The
are x,
a right
right triangle
triangle are
2, and
and x + 5.
5. Which
Which of the
x -‐ 2,
the following
following
equations could
could be
used to find x ??
equations
be used
A)
+X - 2 = X +5
+ (x + 5)2 = (x - 2)2
+ (x - 2)2 = (x + 5)2
(x - 2) 2 + (x + 5)2 = x2
X
square of side
side length
length 6
A square
the figure
6 is shown
shown in the
figure
above. What
the value
above.
What is the
value of x ??
B) x2
C) x2
D)
6
A) WE
A)
3v'2
B) 6
C)
WE
C) 6v'2
D) 6v'3
6\/.’§
B
B
I
C
A
D
Note:
Note: Figure
n o t drawn
drawn to scale.
Figure not
ln
BDC above,
the length
In .6
ABDC
above, what
what is the
length of W
DC??
C
A) 3
the figure
figure above,
above, E
In the
|| CB.
What is the
AB II
CD. What
the length
length
of AB
AB??
B) 5
C) 5\/§
C)
5v'3
D) 8
199
CHAPTER 22 TRIANGLES
CHAPTER
TRIANGLES
Two angles
same measure.
measure.
angles of a
a triangle
triangle have
have the same
IfIf two
15and
the
two sides
sides have
have lengths
lengths 15
and 20, what
what is
is the
greatest possible
greatest
value of the
of
the
possible value
the perimeter
perimeter of the
triangle?
triangle?
Note:
drawn to
scale..
Note: Figure
Figure not
not drawn
to scale
In the
the figure
figure above,
above, the
the base
base of
of a
a cone
cone has
has a
a
radius
sliced
horizontally
so
radius of 6. The
The cone
cone is
is sliced horizontally so
that the top
that
piece
is
a
smaller
cone
with
a
height
top piece
smaller cone with height
of 1 and
and a base
base radius
radius of 2. What
What is the
the height
height of
of
.
the bottom
the
bottom piece?
piece?
N
N
~
5
M
5
8
A) 1
B) 2
C) 3
O
0
D) 4
What
area of isosceles
What is the area
M NO
isosceles triangle
triangle MNO
above?
above?
A
B
H
10
C
Note : Figure not drawn to scale .
the figure
In the
figure above,
above, E
AB is parallel
parallel to
to flGH and
and 57
OF
is parallel
‐C". If
= 1,
1,EH
= 3,EC
2 2,
2, and
parallel to 'BBC.
If DE
DE =
EH =
3, EG =
and
HC =
10, what
what is
is the
length OfAD
of AD??
HC
2 10’
the length
the figure
figure above,
In the
anequilateral
on
above , an
equilateral triangle
triangle sits on
top of a
a square.
top
If the
an
area
of
4,
square. If
the square
square has
has an area
what is the
what
the area
area of the
the equilateral
equilateral triangle?
triangle?
A)) \ v'3
A
/§
\/§
B) )v'3
B
T
2
3
OZ
C) 4
D
D) ) 1
200
THE COLLEGE
COLLEGE PANDA
PANDA
THE
A calculator is allowed
allowed on the following
following
questions
questions..
cC
DE
How
radians are
H
o w many
many radians
are in 225° ??
A)
37T
37r
-
D...,______
T4
7rr
77r
B) -
?6
5rr
Sn
C) -
A
74
0)
E
B
ABC
above, L
CDE =
and AA
L A = 90°
In 6
AA
B C above,
ACDE
= 90° and
90°..
AB
: 9and
DE = 6,
6,whatis
AB =
9 and AC
AC = 12. If DE
what is the
the
length of CE
length
CE??
337r
rr
-
72
A)) 6
A
B) 8
C) 9
0)) 110
D
0
A
~
B
9
C
E
F
w
W
Triangle ABC
ABC above
similar to
to triangle
triangle DEF.
DEF.
Triangle
above is similar
What
is
the
perimeter
of
triangle
DEF
?
What the perimeter
triang le DEF?
A ) 220
0
A)
zZ
20
20
B) 26.8
C
C)) 3300
12
12
D) 36.2
0)
________________
X
J
15
15
yY
Two
above are
are
Two poles
poles represented
represented by
by W
XW and
and W
YZ above
15 feet
20 feet
15
feet apart.
apart. One
One is 20
feet tall
tall and
and the
the other
other is
12 feet
rope joins
joins the
12
feet tall.
tall. A rope
the top
top of one
one pole
pole to the
the
top
of
the
other.
What
is
the
length
of
the
rope?
top
the other . What the leng th
the rope?
ABC,, E
shortest side
side..
In isosceles
isosceles triangle
triangle ABC
BC is the
the shortest
If the
degree
measure
of
LA
is
a
multiple
of
10,
the degree measure
L A is a multiple
10,
what is the
the sma
smallest
possible measure
measure of L
[BB ??
what
llest possible
A) 75°
A
2
A)) 112
B)
B) 70°
B)
B) 17
17
C) 65°
C)
C) 18
18
D) 60°
0)
D)
0) 19
19
201
201
CHAPTER 22
TRIANGLES
CHAPTER
22 TRIANGLES
y
A( - 2,4)
24
35
What is the
What
the perimeter
perimeter of the
the trapezoid
trapezoid above?
above?
A) 100
A)
B) 108
C) 112
D) 116
B(- 2, - 3)
C(5, - 3)
Points A, B, and C form a triangle in the
xy-plane shown above. What is the measure, in
radians, of angle BAC?
A);6
7[
A)
B)?
B) 4
C)?
C) 3
8
7[
7[
X
D);
D) 2
7[
What is the
the value
triangle above?
What
value of x in the
the triangle
above?
y
A
z
D
parallel lines
lines are
Two parallel
are shown
shown in the
xy-plane
the xy-plane
above. If
If AB
AB := 15
and point
point B has
has coordinates
coordinates
above.
15 and
n ), what
what is the
the value
(m, n),
value of n?
n?
ln the
the figure
figure above,
above, ABC
ABCDD is aa square
In
square of side
side
length 3.
AW =
length
3. If AW
: AZ
AZ = CX = CY
: 1,
1 , what
w h a t is
is
CY =
the perimeter
rectangle WXYZ
the
perimeter of rectangle
WXYZ? ?
A
A)) -‐ 66
A
s fi
A) ) 3v2
B)
B) 4\/§
4v2
C
e fi
C) ) 6v2
B
B)) -‐ 8
C)) -‐ 9
C
D
2
D)) -‐ 112
D) 8
202
THE COLLEGE
PANDA
THE
COLLEGE PANDA
B
A
B
D
C
In the figure above, circle O is inscribed in the
square ABCD. If BO = 2, what is the area of the
circle?
In the figure above, equilateral triangle A BC is
inscribed in circle D. What is the measure, in
radians , of angle ADB?
A)
A)
7T
A)
4
7T
27T
27r
?3
37T
37r
8)
B) -
8)
2
C)
7T
D)
T4
37T
2
47T
471
C) C)
“5‑
5
57T
Sn
D)
D)
76
y
D
A
X
2
In the
the figure
figure above,
[n
above, the
the value
value of %
where
~~ is k, where
.
C
constant. Which
Which of the
the following
following ratios
k is a constant.
ratios has
has
1
a
a value
value of ik??
45°
B
What
the length
length of DB
the figure
figure above?
What is the
DB in the
above?
A)
2v'3
M5
3
3
8)
B)
2./6
216
C)
C)
4./6
sfi
3
3
z
Z
3
YZ
A)
A)
flxz
8)
B)
W
xw
C)
C)
W
XY
xy
XY
yz
YZ
YW
YW
D)
D)
XW
xw
\/5
D) v'3
203
CHAPTER 22 TRIANGLES
CHAPTER
TRIANGLES
y
B
B(2,8 )
E
F
+---+--------'~--x
0
A
D
C
0 (8,0 )
In the xy-plane above, points A and C lie on OB
and BO, respectively . If AC is parallel to the
x-axis and has a length of 3, what is the length of
Equilateral triangle
triangle DEF
DEF is inscribed
inscribed in __
Equilateral
equilateral triangle
triangle ABC
ABC such
that E
equilateral
such that
1 AC.
ED 1AC.
What is the
the ratio
ratio of the
the area
area of ADEF
6. 0EF to the
the area
area
What
ABC??
of ABC
BC?
A)) 11 ::4
A
4
B)) 1 ::33
B
1:2
C) 1
:2
0)) 5 ::88
D
v
B
C
A
D
A
A‘
B
the figure
figure above,
above, equilateral
equilateral triangle
triangle AED
In the
AED is
the figure
figure above,
above, a
a semicircle
semicircle sits
sits on
on top
top of a
In the
a
square of side
side 6. Point
Point A is at the
the3p
top of the
the
square
semicircle . What
What is the
the length
length of AB?
semicircle.
AB ?
contained within
within square
square ABCD
the
contained
ABC D.. What
What is the
degree measure
measure of [LBBEC?
degree
BC?
A)
B)
C)
D)
60°
100°
100°
120°
120°
150°
A)
A) 3\/§
3 v15
B) 7
C) 9
D
3m
D)) 3v'IO
204
THE COLLEGE
COLLEGE PANDA
THE
PANDA
In AABC,
ZABC
= 120°
120°..
6 ABC, AB
AB = BC
BC := 6and
6 and L
ABC =
What is the
the area
area of AABC
6 ABC??
What
A
A
N5
A)) 2v'3
B)
B) 4\/§
4v'3
C)
6v'3
C ) s fi
D)
W3
D) 9v'3
3
0
O
4
E
C
In the
square DBCE
DBCE has
has aa side
side
the figure
figure above,
above, square
length
4, what
is
the
length
AD ?
length of 3. If OE
OE = 4,
what the length of
of AD?
y
3!
e
x
( v'3, -‐ 11))
N3,
A
In the
xy-plane above,
above, angle
formed by
the xy-plane
angle 9
0 is
is formed
by the
the
x-axis and
and the
the line
shown.. What
line segment
seg ment shown
What is the
measure, in radians,
measure,
radians, of angle
angle 0 ?
?
?3
B)
B)
T4
-‘ 4
4
F
571
57T
A)
A)
B
12
12
77r
77T
97r
9n
C) C)
?5
1171
lln
D)
D)
T6
D
C
Square
ABC D above
Square ABCD
above has aa side
side length
length of 12. If
If
BF
: 4,
4, what
what is
is the
length
of
BE?
BF =
the length of BE?
A
A)) 3
B)) 2,/2
B
zfz
C)
C) 3\/§
3,/2
D) ) 4,/2
D
Ni
205
CHAPTER
TRIANGLES
CHAPTER22
22 TRIANGLES
A
R
E
D
B
Q
15
15
T
In the
above, RT
RT =
= 17
17 and
the figure
figure above,
and g
QS is
perpendicular
perpendicular to RT.
RT . What
What is the
the length
length of B7"
ST to
C
the nearest
nearest tenth
the
tenth of a unit?
unit?
In the
the figure
figure above,
= 12, AC = 13, and
and
above, AB =
DE = 3. What
is the
DE
What is
the length
length of AE
AE??
A) 12.6
A)
B) 12.8
C) 13.2
D) 13.4
B
C
X
D
2
B~---------JE
4
A
X
A
F
C
Note: Figure
Figure nnot
Note:
o t drawn
drawn to scale.
scale.
Note: Figure
Figure nnot
Note:
o t drawn
drawn to scale.
the figure
figure above,
above, 51?
In the
DE is parallel
parallel to K.
AC. The
The
perimeter
12but
no
perimeter of triangle
triangle BDE
BOE is
is at
at least
least 12
but no
more
p, of triangle
triangle ABC
~ore _than
than 16. If
If the
the perimeter,
perimeter, p,
is
one possible
value of p ??
1san
an integer,
mteger, what
what is one
possible value
In the
above, points
lie on A_C
the figure
figure above,
points B amuz"
and E lie
AC
and
OF, respectively
and fi,
respectively,, such
that BEis
to
such that
BE is parallel
parallel to
5.
What is the
x?
CD . What
the value
value of x?
A) Jé
,/6
A)
B
)
z fi
B) 2./2
C) 2\/§
2)3
C)
D) 3
206
23
Circles
Circles
Should Know:
Circle ~acts
Facts Yoµ
You Should
Know:
Area of a
a circle: rrr2
Area
m2
Circumference
Circumference of a
2m
a circle: 2nr
Arc,,Length:
Arc
Length: ~
% ·x 2m
2,rr
!
Area of a3 Sector:
Sector: % x nr
7tr22
,Ar.ea
OR
OR
Or if 09 is in
in radians
radians
Or
OR
érze if 60is
is in radians
radians
½r20
Central angles have
have the
same measure
measure as
a r t s that they
”carve out.”
<;'.fil\tra.Langles
the same
as the
the arcs-that
they "carve
out."
A
‘
B
Many
Many students
students confuse
length with
with arc
measure. The
length is
is the
the actual
actual distance
one would
would travel
travel
oonfuse arc length
arc measure.
The arc
arc length
distance one
along
the circle from
from A to B. Arc
measure
is
the
number
of
degrees
one
turns
through
from
A
to
B.
You
alotrg the
Ate measure
the number
degrees one turns through from A to B. You
can
think of it as
asa
rotation along
along the
the circle
circle from
from A to B. A
A full
full rotation
rotation is
is 360°.
360°.
can think
a rotation
Inscribed angles
angles are
are half
half the
the measure
measure of the
arcs that
that they
Inscribed
the arcs
they ”carve
"carve out."
out."
60°
I
..
.
207
CHAPTER 23 CIRCLES
CHAPTER
CIRCLES
.Oways 90°.
extension of the
the previous
angle
· inscribed
• . · P)
Angles
in aa sewi~
semicircle arei
are always
90°. This is
is just
just an
an extension
previous fact. An angle
inscribed in aa semicircle
carves out half ~,circle,
a circle, or 180°,
180°,which
which means
means the
angle itself
itself is half
half that,
that, or 90°
90°..
"circle~a,tves
the angle
@
. is
Aradiusdrawntoalinetangenttothecimleisperpendicular
to that line:
./
.;,
E}"eirele perpendicular
to that line:
®
‘x
Gen.etal
eq:uation1)£a .circlemthe__
xy-plane:
Germlequationofacircleinthexy-plane:
(x‐h)2+(y‐k)2=r2
',.'.
y31
',"
II
where~{h,
(h,k)
radius.
wh~
fllsisthecenterofthecimleandrisits
-the center.of the-circleand r is i~ ;radius.
EXAMPLE1:
~ i'
~/'
... ~.\,
+ ... ,.
ln
above, the outer
s radius
as the
the inner
cltcle' s. What
the ratio
In ,the
the .fi~
figure above,
outer citcle'
circle’s
radius is twice
twice as
as long
long as
inner circle’s.
What is the
ratio of the
the
areaoftheshadedregiontotheareaoftheunshaded
region?
of the unshaded region?
~a
o~ the $haded region-to the area
A);
B)1
B)l
C)2
C)2
D)3
0)3
208
THE COLLEGE
COLLEGE PANDA
PANDA
THE
the radius
radius of
of the inner
inner circle
circle be
Then the
the radius
radius of the
is 2r.
Let the
be r. Then
the outer
outer circle is
Area of inner
inner circle: n:r
m22
Area
Area of outer
outer circle: n:(2r
7T(2r)2
)2
Area
of
shaded
region:
7r(2r)2
7rr22 =
: 4n:r
47'tr22 -‐ n:r
m22 =
: 3n:r2
37rr2
Area shaded region: n:(2r) 2 -‐ n:r
Shaded
3n:r2
Shaded
37tr2
3
-U
-n-sh
_a
_d_
e_
d
=
_n:_r_
Unshaded : 7rr22 =
‑
The answer
answer is J ((D)
The
D) J..
EXAMFLE2:
mm 2;
C
B
2
Whatistheareaofmeshadedregioninfltefigureabove?
What
is the areaof the shaded region in the figureabove?
Aug‐Vi wig‐Ni
0125‐2
m g - fi
get the
the shaded
shaded area,
area, we
we must
must subtract
subtract the
the area
area of the
the triangle
To get
triangle from the
the area of the
the sector.
45°
45° n:r
_
sector.. -360°
7Tr2 =
‐Area of sector·
.
'360 °
1
7f
_
7r(2
_ -N
2
-81 n(
22z) =
2
8
triangle: Draw
Draw the
the height
height from
from point
point A to base
45‐45‐90 triangle.
triangle. Because
also
Area of triangle:
base CB. This makes
makes a3 45-15-90
Because AC
AC is also
radius,, its
its length
length is 2.
2. Using
the 45-15-90
45‐45-90 triangle
triangle relationship,
relationship, the
aa radius
Using the
the height
height is then
then %
~ = fl.
,/2..
1
1
1
: ébh
Area =
(2)( v'2)=
=
2bh == 2(2)(fi)
x6
v'2
i-
shaded region
: g ‐ flV2
Area of shaded
region =
The answer
answer is ~-(D) .
The
209
CHAPTER 23 CIRCLES
CHAPTER
CIRCLES
EXAMPLE
3:
EXAMPLE3:
A
120°
0
B
Adielewithadiame’wrof’misshowninfitefigureabova IfziAOB =-120°,whatisthelengthofminor
mXE“?
,
,
$2955
0 ·'1107:
011.
,, 3
t.
120°
1200 (2m ) =
w(2flr)
‐
360°
1
1071’
~(2n
x 5) = 110n
§(27T x5)‐‑
3
3
I
The
The answer
answer is ~(C) .-
“W134:
C
In ,the
the ngure
figureabove,
A C E is
is inscribed
inscribed in
in 0ircle
circle O:_
0. What
What is the
ln
.,above, [LACB
the measure
measure of
of angle
angle ACB ?
?
A) 15°
15°
B) 30°·
30°
C) 45°
C)45°
D)
60°°
D)60
The measure
arc AB
XE is the
as the
central angle
angle LAAOB,
AC B is
The
measure of minor
minor arc
the same
same as
the measure
measure of central
AOB, 90°. Inscribed
Inscribed angle
angle ACB
§j.
half
half of that,
that, 45°. Answer
Answer (C) .
210
PANDA
COLLEGE PANDA
THE COLLEGE
THE
BXAMPLH5:
EXAMPLE5:
fi‐a+f+@=m
x2- 4x + y 2 + 2y = 31
the ·eirde?
eentet of the'circle?
the center
the ,coordihates · of the
are thecoordinatesof
What are
gh'ren.above.
· is
~
the,~
'i:ltd e&:1,
::of ·.aniflemfm
~uati:on
.:i".i
"f
' Theeqnafiom
the“
xy‘plane
is given
above. What
- 2,-‐ 11))
A}(
A )(‐2:
ij))((-42,1)
8
:1)
- 1,2)
C)()(412)
C
0)(2,
D )(2, ‐- 11))
square
the square
complete the
have to complete
we have
r 2 , we
= r2,
k) 2 =
+ (y ‐- k)2
h)2+
form (x ‐- h)2
the standard
circle in the
the circle
To get
equation of the
standard form
the equation
get the
review
should review
you should
square, you
the square,
complete the
how to complete
know how
don't know
the y's.
and once
x 's and
the x’s
once for the
twice, once
twice,
once for the
y’s. If
If you
you don’t
with x,
Starting with
do it. Starting
how to do
contains many
which contains
the
many examples
examples of how
chapter, which
quadratics chapter,
the quadratics
2
((xx-‐ n
++e2y
y ==3311
4 + fy2
-‐ 4
2)2
Then
Theny,y,
u‐2P‐4+@+1F‐1=m
( X - 2 )2- 4 + (y + 1)2- 1 = 31
(x‐n2+@+1V=36
- 2)2 + (y + 1) 2 = 36
(x
Answer ~(D) .radius is 6. Answer
the radius
and the
center is at (2, ‐- 11)) and
can see
we can
standard form,
the standard
From the
From
form, we
see that
that the
the center
211
CHAYTER 23
CHAPTER
23 CIRCLES
CIRCLES
CHAPTER EXERCISE:Answers for this chapter start on page 330.
A calculator is allowed
allowed on the following
following
questions.
questions.
A
B
pP
The circle
circle above
above has
has area
area 367t
36rr and
The
and is divided
divided into
into
congruent regions.
regions . What
What is the
the perimeter
8 congruent
perimeter of
one of these
these regions?
regions?
one
D
C
A)
A) 6+1.57r
6 + l.Srr
B) 6+27T
B)
6 + 2n
In the
figure above,
above, the
the square
[n
the figure
square ABCD
ABC D is
inscribed
in
a
circle
.
If
the
the circle is r,
inscribed a circle. If the radius
radius of the
r,
what is the
the length
length of arc APD
APO in terms
terms of r?
what
r?
A)
12 + l.Srr
C) 12+1.57r
D) 12+27r
D)
12 + 2rr
nr
4
rr;
B
B)) - ‐
Which of the
the following
following is
is an
an equation
a circle
circle
Which
equation of a
the xy-plane
center ( -‐ 22,, O)
0) and
an area
in the
xy‐plane with
with center
and an
area
49rr?7
of 4971
C) nr
C)
D)
D)
nr 2
4
A) ) ((xx- ‐2) 22+) y 2 = 7
A
B)
+y2=7
B) (x+2)2
(x + 2) 2 +
y2 = 7
C) (x
C)
( -‐2)2
+y2=49
2)2+
y2 = 49
D)
+y2=49
D) (x+2)2
(x + 2)2 +
y2= 49
the figure
figure above,
above, three
three congruent
congruent circles
circles are
are
In the
tangent to each
each other
other and
and have
have centers
tangent
centers that
that lie on
on
the diameter
diameter of a larger
larger circle.
the area
the
circle. If
[f the
area of each
each
of these
these small
circles is 97t,
9n, what
what is the
the area
small circles
area of
the
the large
large circle?
A)
A) 3671
36n
B) 497r
49n
C) 64n
647t
D) 8171
81rr
212
THE COLLEGE
COLLEGE PANDA
THE
PANDA
B
B
Note:
Note: Figure
Figure not
not drawn
drawn to scale.
scale.
In
is
ln the
the figure
figure above,
above, equilateral
equilateral triangle
triangle ABC
A BC is
inscribed
D. If
If the
the area
area of
of circ/lxe
D is
inscribed in circle
circle D.
circle Dis
36m
the length
minor arc
36n, what
what is the
length of minor
arc AB
AB ??
In the
CB is inscribed
the figure
figure above,
above, [L AACB
inscribed in aa
-
circle. The
The length
length of minor
circle.
minor arc
AB is what
arc AB
what
fraction of the
the circumference
fraction
the circle?
circle?
circumference of the
A) 27r
2n
A);
1
B) a);
4
C);61
C)
1
A) 3
B) 37r
3n
B)
47T
C) 47r
D)
D) 67r
6n
mg
1
D) 12
C
In the
C has
radius of
of 6.
If
the figure
figure above,
above, circle
circle C
has aa radius
6. If
the
the shaded
shaded sector
1071, what
what is the
the
the area
area of the
sector is 10n,
measure,
AC B ?
measure, in radians,
radians, of angle
angle ACB?
A)
A)
In the
above, AC is a
circle
the figure
figure above,
a diameter
diameter of the
the circle
and the
and
AB is 1.
1. If the
radius of
the length
length of AB
the radius
of the
the
circle is 1, what
circle
degrees, of
what is the
the measure,
measure, in degrees,
LBAC
L BAC??
27r
27T
?5
471
47T
79
57r
57T
C)
79
Sn
57T
D)
D)
T8
B)
B)
213
CHAPTER 23 CIRCLES
CHAPTER
CIRCLES
A
C
B
1n the
the figure
figure above,
above, a circle has
In
center C and
and
has center
radius 5. If
measure of central
radius
If the measure
central angle
angle AC
ACBB is
7T
7T
4
2
The base
base of a right
right circular
circular cylinder
The
cylinder shown
shown
above
has
a
radius of 4.
above has radius
5. What
4. The
The height
height is 5.
What is
surface area
area of the
the cylinder?
cylinder?
the surface
between g and
and g radians,
radians, what
what is one
one possible
between
possible
A) 4
40rr
A)
0”
60rr
B) 607r
integer value
value of the length
integer
length of minor
XE ?
minor arc
arc AB?
72rr
C) 7271
D)
81rr
D) 8171'
H
_...;..._--------~------1
figure above,
above, circle P and
In the figure
and circle U each
each
have
radius of 3 and
and are
are tangent
have a radius
tangent to each
each other.
other.
.6.P HU
equilateral, what
ifIf API-I
U is equilateral,
area of the
what is the area
shaded region?
shaded
region?
1n the figure
In
figure above,
above, four
four circles, each
each with
with radius
radius
are tangent
tangent to each
each other.
other. What
What is the
4, are
the area
area of
the shaded
shaded region?
the
region?
A)) 16
A
1 6-- 44rr
7r
B
)
6
4
‐
4
7
B) 64 - 4rr1
C
7r
C)) 6
644 -‐ 88rr
A) lOrr
A) 1 0 "
B)) 1271
12rr
B
D)
D) 64
64‐167r
- 16rr
D) 1
15rr
D)
5”
C)) 1471
14rr
C
214
THE
COLLEGE PANDA
THE COLLEGE
PANDA
(x + 2)2 + (y + 4)2
=4
The equation
equation of a
a circle
circle in the
the xy-plane
xy-plane is given
The
given
above . Which
Which of the
the following
must be
be true?
above.
following must
true?
I. The
the circle
).
The center
center of the
circle is at (2,4
(2,4).
II.
tangent to the
the x-axis.
I I , The
The circle
circle is tangent
x-axis.
III.
The circle
circle is tangent
tangent to the
the y-axis
I I I , The
y‐axis..
A) I]II only
only
A)
B) III
only
I I I only
and II
only
11only
C) I and
D)) II,, III,
D
I , and
and 111
Ill
AA
Note
Note:: Figure
Figure not
n o t drawn
drawn to scale.
scale.
If
If the
the area
area of the
the shaded
shaded region
region in the
the figure
figure
24rr
and
above
is
the
radius
of
circle
O
above 247r and the radius circle 0 is 6,
6, what
what
is the
the value
value of x ?
A)
A ) 115
5
B)
B) 30
C)
45
C) 45
D)) 6600
D
A
AA
D
B
In the
the figure
figure above,
above, circle
circle A is tangent
tangent to circle
circle B
3
at
the circles
at point
point D. If
If the
circles each
each have
have aa radius
radius of 4
and
AC is tangent
and A‐C
tangent to circle
circle B at
at point
point C, what
what is
the
triangle A
BC ?7
the area
area of triangle
ABC
A) 8
B ) 8y'2
s fi
B)
est/5
C) 8v'3
D ) 116
6
D)
215
24
ometry
Trigonometry
Trigon
“
need to know:
you need
functions you
illustrate the three
triangle to illustrate
We'll use
right triangle
three trigonometric
trigonometric functions
know:
5-12-13 right
a 5‐12‐13
use a
13
5
5~
u
sinxzmzi
5
opposite
.
= sm x = -~--hypotenuse
13
13
hypotenuse
12
12
c o s x z n g
opposite
= -5
tanxzopPOSltezi
tan X = ----'---''--12
adjacent
·
12
adjacent
= cosx = -~-hypotenuse
hypotenuse
13
adjacent
12
they ' re just
were just ordinary
as ifif they were
functions as
trigonometric functions
It’s important
ordinary numbers.
numbers. After
After all, they’re
these trigonometric
important to see these
It's
.
.
.
. .
1
similar.
angle are Similar.
30° angle
a 300
with a
triangles With
right triangles
ratios.
always equal
equal to 5.. Why? Because
Because all right
example, sin 30° is always
ratios . For example,
2
The ratios
ratios of the sides
sides stay the same.
same.
2
1
.
\f3
v'3
I
4
2 v'3
2\/§
tan x differently
and tan
treat sin x, cos x, and
because they treat
Many
students over-complicate
differently than
than regular
regular
trigonometry because
over-complicate trigonometry
Many students
following:
the
like
mistakes
make
sometimes
students
,
numbers. Perhaps
Perhaps because
because of the notation
notation, students sometimes make mistakes
following:
numbers.
.
sin 2x
: ssm
i n2
--=
X
treat them
and treat
2x and
and 2x
sin and
The above
n o t possible
possible because
because sin 2x
o n e "entity."
"entity.” You cannot
them
separate sin
cannot separate
2x is one
above is not
independently just like you
you can't
can’t separate
x ) into
into f and
and x.
separate Jfl (x)
independently
216
THE COLLEGE
COLLEGE PANDA
THE
PANDA
The definitions
definitions of sine,
The
cosine, and
and tangent
tangent are
are best
memorized through
through the
c t m y m SOH-CAH-TOA,
sine, cosine,
best memorized
the aacronym
SOH-CAH-T OA,
S for
over hypotenuse),
S
for sine (opposite over
hypotenuse),C
for cosm'
e (adyacenft over
and T
t (opposite
C for
cosine_(adjacent
ovet hypotenuse),
hypotenuse),and
T for
for tangen.
tangent(opposite
ov~r
w
e t adjacent)
'
t)..
momite
Aside
you should
should also
also memorize
identity:
Aside from
from the
the definitions,
definitions, you
memorize the
the following
following very
very important
important identity:
sinx = cos(90°
cos{90° - x)
sinx
Themeverseisalsotrue.
The reverse is also true.
cosx=sin(90°‐x)
cos x = sin(90 ° - x)
Expressed
in radians,
Expressedin
radians,
sinxzcos(‐72£‐-x)
sinx = cos ( - x)
1
. (1f ) .
cosx=sin(g‐x)
COS X = SlI\ Z - X
and
and
.
Now,
the trig
functions depends
Now, the sign
sign of each
each of the
trig functions
depends on the
the quadrant
quadrant in which
which the
the angle
angle terminates.
terminates.
y
I
II
-----0
-+------
m
X
IV
• Sine,
Sine, cosine,
cosine, and
o
tangent are
all positive
positive in
in the
the first quadrant.
and tangent
are all
quadrant.
0
Only sine
sine is
is positive
podfive in
in the
the second
second quadrant.
quadrant.
• Only
0
tangent is
is positive
the third
third quadrant.
• Only
Only tangent
positive in
in the
quadrant.
• Only
Only cosine
cosine is
0
is positive
positive in the
the fourth
fourth quadrant.
quadrant.
These are
are best
through the
Students Take Calculus).
Calculus). All
are
Those
best memorized
memorized through
the acronym
acronym ASTC
ASTC (All Students
A l l the
the functions
functions am
positive
quadrant, only
only sine
the second,
second, and
positivein
in the
the first
first quadrant,
sine is
is positive
positive in the
so
on.
and so on.
217
CHAPTER24
CHAPTER
24 TRIGONOMETRY
TRIGONOMETRY
' 'I;Q.
find the
tb:e~,µ~e
a qijg p;mdion
:without a'oalculator,
a ·ca1culator, .
Tofin'd
value of
ofa’ttig
Extraction'for-an
foran angle
angle without
J)et ~tiiune w:hatth~cSignof the .results'he.tild
be (positive ornegative).
or negative).
' >1.:
la.mewhat&iesigxioffiiemlt
shouldbe
.
.2(
referen~~gle (the
drawing aa straight
straight line
X'-axis). If
the
Z Fin,d.the
Find flievzefiexenoemgle
(the acute angle you
you get
get by drawing
line to the
the x‐axis).
If the
angle
·is 225°, for example, th reference angle is 225° -‐ 180° =45°:
= 45°.:
angleis225°,f01'example,tchenel‘erenceanglei5225o
·,
'+
l •
..
"'-. ' ,,.
·"' .. ,·
•
...
,\1,.'' ',· ..
J-1•'
'
...
t
l
•
''
,
:
t .,
,t
~--.i..~•
/'
y
~
"'
:
. .
l~ , l
'I
,1 ••~
1
:. }
~
•
,c,- ..
2250
•/
,1,,,t
:t;,1
Don’t tQemorize
memorize any
formulas £bi;
for finding the
the reference
reference angle.
draw aa line
line to -the
the x-axis
x-axis and
Dontt
arty formulas
angle. Inst
Just draw
and
fig:u:.re
if out yow:se:lf!
figmeitoutyomselfl
3. Use
Useyourworm‐Mspecialfighttinglestogetthetfigvaluefortherefemceangle.
3.
your 45-45--90.or 30-60-,90.special right ttiangles to get the trig value for the refe:r;enceangle.
l'he SAT won't
to calculate trig
values for
angles that
aren't in thesespecial
these special right
TheSAT
won't as~
ask you
you tocalculate
trigvalues
for angles
that aren'tin
right triangles
triangles
~ess
ryou'
re
able
t-0
use
your
calculator,.
unless-ymflmabletouseymcalculator.
4. Makesumyourresnlthasthecomectsignfimstepme.
M:akE}sw:e
your result has t:l,;le
c;o,rect sign from step one.
4.
Let's do
Let’s
do aa couple
couple simple
simple examples
examples..
l.
1. What
What is the
the value
value of sin
sin 330°?
fourth quadrant
quadrant and
and sine
sine is negative
negative in the
quadrant, the
the result
should be
be
Since 330° is in the fourth
the fourth
fourth quadrant,
result should
the reference
help of a
a diagram:
diagram:
negative
negative.. Now
N o w let'
let’ss find
find the
reference angle
angle with
w i t h the
the help
y
330°
The reference angle is 360° - 330°
= 30°. Using the 30-60-90
1
v'3
. 300 opp
1
= = hyp
2
Stn
I-~/.
Since the result
result should
should be
be negative,
negative, sin
sin 330° = ‐12 .
218
218
triangle ,
THE
THE COLLEGE
COLLEGE PANDA
PANDA
2. What
the value
What is the
value of cos 135°7
135°?
Since
the second
second quadrant
quadrant and
and cosine
cosine is negative
second quadrant,
Since 135° is in the
negative in the
the second
quadrant, the
the result
result should
should be
be
negative . Next,
negative.
Next, we
we find
find the
the reference
reference angle
angle::
y
135°
reference angle
angle is 180°
180° -‐ 135° = 45°. Using
Using the
the 45-4545‐45‐90
The reference
90 triangle,
triangle,
1
1
cos 45°
=
= - 1- = v'2
adj
hyp
,/2
2
1-1;1.-
Since the
the result
result should
should be
be negative,
negative, cos 135° =
: ‐ ‐‐
3. What
What is the value
value of tan
tan 210°7
210°?
Since 210° is in the
positive in the
the third
third quadrant,
quadrant, the
the result
result should
the third
third quadrant
quadrant and
and tangent
tangent is positive
should be
be
positive . Next,
positive.
Next, we
we find
find the reference
reference angle
angle::
y
210°
The reference
the 30--60
- 90 triangle
triangle shown
shown earlier,
earlier,
reference angle
angle is 210° -‐ 180° = 30°. Using
Using the
30‐60‐90
\/§
0
11
)3
t a n 3=0 -° = =‐ -= ‑
tan30
\/5
)3
J;
\/3
result should
should be
be positive,
tan 210° =
= I ‐3‐ 1.Since the result
positive, tan
219
3
3
CHAPTER 24 TRIGONOMETRY
CHAPTER
TRIGONOMETRY
• j•
Finall~
Myywshmldnmoflzeflxefofloufingvahzesformmdw:
you should memorize the following values for 0° and 90°:
'
~
sin0°=0
sin0
° =0
cosO°=1
cos0 ° = 1
tan0°=0
tan.O° = 0
sin90°=lV
sin90 °= 1
~-,
cos90°=0
cos90° = O
=
m190°=undefined
tan
90° undefined
Expmsshagwmmdians,
fag,ressing 90° in radians ,
(i) =_1
cos (i) = 0
sin
t¥t (;) = undefined
220
220
•
t l
"'i
.,•.¢
,,...,
1
.,.t •• -
;,
,I
I,
THE COLLEGE
COLLEGE PANDA
THE
PANDA
CHAPTER EXERCISE: Answers for this chapter start on page 332.
calculator should
A calculator
O T be
should N
NOT
be used
used on the
the
following
questions.
following questions.
In right
is 90°
right triangle
triangle ABC,
ABC, the
the measure
measure of AC
L'.C is
and
If cosA
2, what
and AB
AB =
= 30. If
cos A := ~,
what is
is the
the length
length of
of
If
If cos 40° = a, what
what is sin 50° in terms
terms of a
a??
AC??
AC
A) a
1
B) -
a
E
C) 9
900-‐ a
D
a fi
D) ) a../2
Iftanx
m , wha
w h att is
i s sin
s i nxx iinn tterms
e r m sof
o fm
m?
1f
tan x =
= m,
In a
a right
right triangle,
triangle, one
such
one angle
angle measures
measures x°
x 0 such
0
that tan
tan x°
x := 0.75. What
that
the value
What is the
value of cos xx°0 ??
1
A
A)
*
) x/m2+1
Jm + 1
2
B
B) )
1
;
C
C)
rn
m
‐
_
m
‐
J1 - m2
\/1‐m2
) Vm2+l
Jm2 + 1
D
D)
) x/l‐m2
J1 - m2
sin 0 + cos(90
cos (90 -‐ 0)
sinG
6) + cosbl
sin(90
6)
cos 0 + sin
(90 -‐ 0)
For any
For
any angle
angle 9,
0, which
which of the
the following
following is
equivalent to the
the expression
equivalent
expression above?
above?
B
A) 0
B) 22sin
sin 6
0
C) 2
cos 0
2cos
e
5
D) 2(sin
+ cos 9)
2(sin 0 +
0)
A
C
Given that
Given
: 5 and
right
that AB =
and tan
tan B = ;g in the
the right
triangle above,
triangle
what is the
the value
value of
above, what
sin
+ cos B
sin B +
B??
221
TRIGONOMETRY
CHAPTER
CHAPTER 24 TRIGONOMETRY
following
allowed on the following
A calculator is allowed
questions.
questions.
- 12)
(Sm ‐cos32
sin(5m
12)
cos32 = sin
are in
measures are
angle measures
In the
the angle
above, the
equation above,
the equation
what is the
m < 90°, what
degrees.
the value
value
If 0° < m
degrees. If
m??
of m
B
3
I
A
C
C
the
BC in the
length of BCin
the length
what is the
If
x 0 = 0.25, what
sin x°
If sin
above?
triangle above?
triangle
y
A( - 3, 5)
N
B
B
A
CM
C M
C(12, - 3)
lane
the xy-p
shown in the
Right
xy-plane
ABC is shown
triangle ABC
Right triangle
va lue of cos C ?
above.
the value
What is the
above . What
0
A
A))
similar
triangle ABC
right triangle
In the
ABC is similar
above, right
figure above,
the figure
In
and C
B, and
A, B,
vertices A,
with vertices
to right
N O , with
MNO,
triangle M
right triangle
0,
and 0,
corresponding
vertices M, N, and
corresponding to vertices
value of
the value
what is the
B = 2.4, what
tan 8
If tan
respectively.
respectively . If
cos
N ?
cosN?
B)
8
_8
17
17
i8
15
13
C) E
15
D)1‐7
222
THE
PANDA
THE COLLEGE
COLLEGE PANDA
B
A
Note:
Note: Figure
o t drawn
scale.
Figure n
not
drawn to scale.
:
In the
the figure
figure above,
abo:7e, cos(90°
cos(90 ° -‐ x°)
x0 ) =
8
it;
is
. What
What is
17
C
In
cos x°
: ~g.
In right
right triangle
triangle ABC
ABC above,
above, cos
x0 =
If
the value
value of
If BC := NE,
2 /k, what
what is the
of k
k?7
the value
value of cos xx°0 ??
the
A)
A)
6
E8
15
15
17
17
B)
E
B) 15
C) 8
8fi
C)
17
15
D) 15
D)
fi
17
B
In a
a right
right triangle,
triangle, the
the sine
the two
sine of one
one of
of the
two
J;.
acute angles
angles is ?.
acute
What
other
What is the
the sine
sine of the
the other
acute
acute angle?
angle?
1
A)
A) 5
2
B)
A
C
Note:
o t drawn
Note: Figure
Figure n
not
drawn to
to scale.
scale.
j3
2
In the
the figure
figure above,
above, ABC and
and DBE
DBE are
are right
right
triangles.
10and
triangles . If
If DE =
= 10
and the
the tangent
tangent of
of angle
angle
BAC is 1.25, what
of segment
what is the
the length
length of
segment BE
BE ??
1
C)
C)
/3
D)
D)
,Ii
2
223
CHAPTER 24
CHAPTER
24 TRIGONOMETRY
TRIGONOMETRY
the figure
figure above,
above, AC
In the
AC is a
a diameter
diameter of the
the
AC := 1,
1, which
circle. If AC
gives
which of the
the following
following gives
the area
area of triangle
triangle ABC
the
ABC in terms
terms of 0
0??
0
A) 5
A)
2
tan0
m
m)
B) ) ‐ 2
B
2
2sin0
C) ZsinG
D)
D)
sin0cos0
sin9c056
2
Given that
that sin
sin 9
0 ‐- cos 0 =
Given
= 0, where
where 9
0 is the
the
radian measure
measure of an angle,
radian
of
the
angle, which
which
the
following could
following
could be
be true?
true?
7T
rr
I,0<9<'§
I.O
< 0< 2
rr
II. g << e0 << 7r
11.
rr
2
377
3rr
Ill. H
rr < 9
0< ‑
III.
22
A)
A) [only
I only
B)
B) II
II only
only
C)
C) I and
I I only
and IIII
only
D)) I,
and III
D
I , II,
I I , and
III
224
25
Data
Reading
Reading Data
easiest
are typically
these are
Fortunately, these
charts . Fortunately,
and charts.
The SAT
your ability
ability to read
read graphs
graphs and
typically the easiest
loves to test your
SAT loves
the
with
arithmetic
simple
on
you
test
just
them
of
Most
math.
much
too
involve
never
they
questions
much math. Most them just
you
simple arithmetic with the
because they never involve
questions because
graph. Practice
a graph.
extra
having to interpret
interpret a
Practice away!
away!
step of having
extra step
225
CHAPTER 25 READING DATA
CHAPTER EXERCISE:Answers for this chapter start on page 334.
following
allowed on the following
A calculator is allowed
questions.
questions .
Voter Turnout
Turnout in
Elections
Congressional
Presidential Elections
and Presidential
Congressional and
80
75
70
E.
....
Cl) 70
.....
~
0 65
> 65
cii
60
.....60
0
.....
55
55
.....
0
.....50
c::
Cl) 45
45 m.‘ __
~
Cl)
40
0... 40
35
30
Commute Times
90
c,
..
-" 75
1/)
~
E 60
_g
<II
E 45
~
B•
~
::, 30
E
E
D•
(3 15
15
30
45
60
75
Commute
Time to Work
Commute Ttme
75
_ l ' _ -+. _ Congressional Election 1
........
Election |
Presidential Election
+
Presidential
I
"'...
90
plotted the commute
For
commute
days, Alex plotted
work days,
four work
For four
work in
time from work
commute time
the commute
and the
time
time to work and
was
days was
which of the four days
above. For which
the
grid above.
the grid
work the
and from work
the
commute time to and
total commute
the total
greatest?
greatest?
---,,.~
-
--
--- ...__
---
-~~
I
~ ~
~ ~
Year
8
N
;::
~ ~
~ ~
~
turnout for
voter turnout
the voter
The graph
shows the
above shows
graph above
election or aa
congressional election
a congressional
each
year a
each year
which ttwo
held. In which
was held.
election was
presidential
wo
presidential election
turnout
voter turnout
difference in voter
year
was the difference
period was
year period
and the
between
election and
congressional election
the congressional
between the
smallest?
presidential
election the smallest?
presidential election
A
A)) A
B) B
A)
A} 1996 to 1998
C
C)) C
B) 2000 to 2002
D
D
D)) D
C) 2004 to 2006
D) 2008 to 2010
226
PANDA
THE COLLEGE
COLLEGE PANDA
Cream Sales
Sales
Ice Cream
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monthly
the monthly
shows the
above shows
The line
graph above
line graph
According
year . According
last year.
Kathmandu last
precipitation
precipitation in Kathmandu
September
precipitation in September
total precipitation
the total
graph, the
to the graph,
precipitation in
total precipitation
the total
percentage of the
what percentage
was
was what
June?
June?
Q\\0\'?f’§C}
cream
above, ice cream
graph above,
line graph
the line
According to the
According
and in 2014
both in 2013 and
highest both
were highest
sales
sales were
period?
three month
during which
during
which three
month period?
A) 40%
8) 50%
B)
March
A) January
January to March
June
to June
April to
B) April
B)
C) July to September
September
C) 60%
D) 75%
December
October to December
D) October
227
CHAPTE R 25 READING
READING DATA
CHAPTER
DATA
Population in
Population
in 2010
Birth Rate
Rate
Birth
-40 1-E1--.;::--c------~
-+-
South Korea
Japan
San
San Diego
Diego
Chicago
Chicago
......
~
1! 20 1---------
----
~H
co
10
10 2006
1
Population (millions)
2006
Researchers
above to compare
Researchers created
created the
the graph
graph above
compare
their
population
estimates
with
the
actual
their population estimates with
actual
populations of different
different cities in 2010. For which
populations
which
the cities did
did the
researcher s underestimate
underestimate
of the
the researchers
the population?
population?
the
2008
2010
Year
2012
2014
Based on
on the graph,
graph, which
which of the
the following
following best
best
describes
the
general
trend in birth
birth rates
rates in
describes
general trend
South Korea
Korea and
and Japan
South
Japan from
from 2006 to 2014?
Each year,
birth rates
rates decreased
A) Each
year, birth
decreased in both
both
South
Korea and
South Korea
and Japan
Japan..
Each year,
both
B) Each
year, birth
birth rates increased
increased in both
South Korea and
Japan .
South
and Japan.
birth rates
rates increased
increased in South
South
C) Each year, birth
but decreased
decreased in Japan.
Japan .
Korea but
Each year, birth
birth rates
rates decreased
South
D) Each
decreased in South
but increased
Korea but
Japan..
increased in Japan
I. San
San Diego
Diego
1,
II. Chicago
Chicago
11,
Ill
111.. Los Angeles
Angeles
A)
B)
C)
D)
D)
~"-c-
t::
Angeles
Los Angeles
I only
and 11
onl y
lI and
11only
lI and
III only
II
and 111
only
I,
and IIII
l, II,
11, and
II
228
228
PANDA
COLLEGE PANDA
THE COLLEGE
I
30
28
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4
5
Hours since 9:00 A.M.
the
created the
researchers created
study, researchers
certain study,
a certain
In a
the
the ages
above to summarize
scatterplot
scatterplot above
Summarize the
ages of the
hours of sleep
the number
and the
participants and
participants
number of hours
sleep
following
the following
Which of the
night. Whichof
each night.
required each
they required
age, in years,
the age,
the closest
is the
closest to the
years, of the
the
least amount
required the least
who required
participant who
participant
amount of
night?
each night?
sleep each
sleep
up
picks up
Musa picks
day, Musa
each day,
A.M.
Starting at 9:00 A
Starting
M . each
trailer
until his
locations until
various locations
packages at
packages
at various
his trailer
then
maximum capacity.
its maximum
reaches its
truck reaches
truck
capacity. He
He then
that
up
picked
he
that
packages that he picked up that
the packages
all the
delivers
delivers all
his
of
weight
the
shows the weight his
above shows
The graph
day . The
day.
graph above
during the
points during
different points
at different
truck at
truck
the day.
day. What
What is
hold, in
can hold,
truck can
Musa's truck
weight Musa's
maximum weight
the maximum
the
tons?
A)
A ) 335
5
B) 40
A
4
A)) 114
55
C) 55
C)
B)
16
8) 16
D
D)) 6600
C
4
C)) 224
D
D)) 3300
229
CHAPTER 25 READING
READING DATA
CHAPTER
DATA
Annual Salt Production in the U.S.
Video Game
Video
Game Console
Console Sales in 2015
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2009 2010 2011 2012 2013 2014 2015
Year
A
B) 2012 to 2013
B)
C) 2013 to 2014
A)
A) A
D) 2014 to 2015
D)
B) B
C) D
D
D) E
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200
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Wolf
Goat
Cat
Cat
D
E
The graph
graph above
the number
number of units
sold
The
above shows
shows the
units sold
in 2015 for five different
different video
game consoles.
video game
consoles.
The
prices of consoles
consoles A, B,C,
B, C, D, and
The prices
are $100,
and E are
and $300, respectively.
respectively. Which
Which
$150, $200, $250, and
of the five consoles
consoles generated
generated the
the most
most total
total
revenue?
revenue?
A) 2009 to 2010
......
C
Console
Based on the
the graph
graph above,
above , for which
which of the
Based
the
following ttwo
consecutive years
years was
was the
the percent
percent
following
w o consecutive
increase in U.S.
annual salt
production the
the same
increase
U S . annual
salt production
same
as the
the percent
percent decrease
decrease from
from 2010 to 2011?
as
ro
M
..0
B
Pig
Animal
According to the
above, the
According
the graph
graph above,
the average
mass
average mass
of a wolf's
wolf's brain
brain is what
what fraction
fraction of the
the average
average
mass of aa pig’s
pig's brain
mass
brain??
230
IBE
COLLEGE PANDA
THE COLLEGE
PANDA
20
,....,_
...
Ill
State Health
Care Spending
State
Health Care
Spending in 2013
-e-CompanyX
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3
0
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AL
Quarter
Quarter
AK
AZ
AR
State
The graph
shows the
the profit
profit of Company
Company X
The
graph above
above shows
and
Company Y in each
each quarter
and Company
quarter of last
last year.
year. In
which
which quarter
quarter was
was Company
Company X's
X’s profit
profit twice
twice
Company
Company Y's?
Y’s?
The graph
graph above
shows the
health care
spending
The
above shows
the health
care spending
four different
different states,
of four
states, Alabama
Alabama (AL),
(AL), AK
(Alaska),
(Arizona), and
and AR (Arkansas)
(Alaska), AZ (Arizona),
(Arkansas) in
the graph,
had the
2013. Based
Based on
on the
graph, which
which state
state had
the
highest combined
and prescription
highest
combined hospital
hospital care
care and
prescription
drug
spending in 2013?
drug spending
A) 1
B) 2
A) Alabama
Alabama
C) 3
B) Alaska
Alaska
D) 4
C)
Arizona
C) Arizona
D) Arkansas
Arkansas
231
CHAPTER
DATA
CHAPTER 25 READING
READING DATA
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1
2
3
4
5
6
180 -170
160
150
J
140
130
120
110
0
Time (hours)
'
i\
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7
,_ --
I
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i
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'
1
2
..
3
4
5
6
7
~ ._ ..
~
8
9
Time (Hours after 8:00 A.M.)
Jeremy
Jeremy works
works at aa call
call center.
center. The
The graph
graph above
above
shows
average number
shows the
the average
number of calls he
he answered
answered
per hour
per
hour during
during his
his 7-hour
7-hour work
work shift.
shift. What
What is
is
the total
total number
answered during
the
number of calls he
he answered
during his
his
shift?
shift?
the day
day of a medical
Greg ate
ate
On the
medical evaluation,
evaluation, Greg
breakfast at 8:00 A.M. and
and lunch
at 12:00 PM.
P.M.
breakfast
lunch at
During each
each meal
doctors recorded
During
meal,, doctors
recorded his
his glucose
glucose
levels in the
the graph
graph above
above until
until they
they were
able to
levels
were able
calculate the
the glucose
glucose recovery
the time
time it took
took
calculate
recovery time, the
the body's
body 's glucose
glucose level
level to return
return to its
initial
for the
its initial
value at
at the
the start
start of the
the meal.
meal. According
According to the
the
value
graph, by how
how many
many hours
hours was
Greg's glucose
glucose
graph,
w a s Greg's
recover y time
time after
after lunch
greater than
than his
recovery
lunch greater
his
glucose
recovery
time
after
breakfast?
glucose recovery time after breakfast?
A) 1.5
B) 2
C) 3
D) 5.5
232
THE COLLEGE
COLLEGE PANDA
PANDA
THE
Car X
§
55
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45
40
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10
20
30
40
50
60
70
Speed (miles per hour)
mileage for Car
The
above shows
shows the
the gas
gas mileage
Car
graph above
The graph
how
graph,
the
on
Based
.
X at
different
speeds.
Based
on
the
graph,
how
speeds
at different
Car X
drive Car
many
gallons of gas
gas are
are needed
X
needed to drive
many gallons
per
miles
30
of
speed
for 5 hours
hours at a constant
miles per
constant speed
hour?
hour?
233
26
Probability
Probability
Generally speaking,
speaking, probability
probability can
can be
be defined
defined as
as
Generally
number of target
target outcomes
outcomes
number
number of total
total possible
possible outcomes
number
outcomes
Nearly all probability
probability questions
questions on
on the
the SAT will
will involve
involve tables
80 for the
purposes of the
Nearly
tables of data.
data. So
the purposes
the SAT,
probability can
can more
m o r e narrowly
narrowly be
be defined
defined as
as
probability
‐-‑
number in target
target group
number
group
number in group
group under
under consideration
number
consideration
EXAMPLE 1:
EXAMPLE
Beef Oricken
First Class
Coach
18
27
62
138
The
a first class
The table
table above
above summarizes
summarizes the
the meat
meat preferences
preferences of passengers
passengers on aa particular
particular flight.
flight. If
If a
passenger
passenger is
is chosen
chosen .at
at random
random from
from this
this flight,
flight, what
what is
is the
the probability
probability that
that the
the passenger
passenger chosen
chosen prefers
prefers
hoofl
·
beef?
9
A)~
POE
40
2
B)
~
B)§
5
3
C)
C)§~
5
D)~
D)§2
3
number of first class passengers
passengers is 18
18 + 27 =
: 45. This
This is the
the group
under consideration.
consideration. The
The number
number of
The number
group under
class passengers
passengers who
who prefer
prefer beef
beef is 18. This
This is the
the target
target group.
group.
first class
number in target
target group
group
number
number in group
group under
under consideration
consideration
number
Answer (B) .
Answerefm.
234
234
18 2
18
45 =
T 5
45
‐-‐‑
THE
THE COLLEGE
COLLEGE PANDA
PANDA
EXAMPLE
manager of a large assembly line uses the table below to keep track of the number of
m m . 2:- The
Themanagerofalargeassemblylineusesthetablebelowtokeeptrackoffliemmmerof
vehicles
are produced
during different
the clay.
day.
vehicles that
that are
produced during
different shifts
shifts in the
Cars
First,shift •
Second shift
Third shift
“m
“Total
173
182
165
520
m
Trucks
-
126
143
109
378
. .:'
Total
299
325
274
898
m
If a vehicle is
closestto
probability
is selected at random at the end of the day, which of the following
following is closest
to the probability
that
the vehicle will be either a car produced during the first shift or a truckproduced duringthethird
during the third
thatthevehiclewillbeeitheracarproduceddufingthefirstshiftoratruckproduced
shift?
A)
0.193
A)0.193~
•B)
B) 0.314
0.314
C) 0.352
(30352
D) 0.421
In this
all the
the vehicles,
vehicles, aa total
total of 898 at
at the
the end
the day.
day.
this question,
question, the
the group
group under
under consideration
consideration includes
includes all
end of the
The
and trucks
trucks produced
during the
the third
The target
target group
group includes
includes cars
cars produced
produced during
during the
the first
first shift
shift and
produced during
third shift,
shift, aa
total
total of 173 +
+ 109 =
= 282 vehicles
vehicles..
number in target
target group
group
number
= 282 ~
314
.
.
.
‐
x 00.314
number
898
·
number in
m group
group under
under consideration
con51deration
Answer (B) .
Answer~.
235
235
CHAPTER
CHAPTER 26 PROBABILITY
PROBABILITY
‑
CHAPTER EXERCISE:
EXERCISE: Answers
Answers for this
this chapter
start on page
chapter start
page 336.
A calculator is allowed
allowed on the following
following questions.
questions.
Speeding
Truck
T
ruck
Car
m
Total
68
83
151
Violation
Violation Type
Stop sign Parking
17
39
51
26
90
43
‐
‐
Total
124
160
‑
284
A district
driving violations
violations by type
vehicle in
above.. According
According
district police
police department
department records
records driving
type and
and vehicle
in the
the table
table above
to the
which of the
the following
is closest
closest to
to the
the proportion
violations committed
committed by
the record,
record, which
following is
proportion of
of stop
stop sign
sign violations
by truck
truck
drivers?
drivers?
A) 0.137
B) 0.315
C) 0.433
D) 0.567
----‑
Color
Color
Red
Red
Blue
Percent
20%
33%
Black
10%
White
White
Silver
14%
car manufacturer
produces cars
cars in red,
red, blue,
blue, black,
A car
manufacturer produces
black, white,
white, and
table above
above shows
shows
and silver.
silver . The
The incomplete
incomplete table
the percentage
the
percentage of cars
cars it produces
each color. If
from the
chosen at
at random
random,, what
what is
produces in each
If a car
car from
the manufacturer
manufacturer is chosen
the probability
probability that
that the
the
silver?
the car’s
car 's color
color is red
red or
or silver?
A) 23%
B) 33%
C) 37%
D) 43%
236
PANDA
COLLEGE PANDA
THE COLLEGE
THE
-------------------T------------------information.
following information.
Questions 3-4 refer to the following
Questions
different
experience in five different
year of experience
one year
least one
with at least
California with
workers in California
number of workers
the number
shows the
below shows
table below
The table
The
occupations.
construction-related occupations.
construction-related
‐‐ ‐ - - -‐ ‐- ‐ ‐ | ‑|
Experience
Years of Experience
Painter
1
2
3
4
5+
Total
22,491
26,973
29,086
33,861
37,061
149,472
‐Roofer - 23,908 -27,634 30,932
- 34,146
‐ 39,718- 156,338 |
169,240
42,680
36,902
32,784
‐Welder - 27,062 -29,812
|
183,885
45,376
40,083
36,670
33,119
28,637
Plumber
‐
- 28,806 - 34,867 ‐ 37,418 - 43,922 - 169,409 |
24,396
Carpenter
‐
‐
|
s2s,344
11
2os,757
1
1s2,410
1
164,339
1
146,344
1
126,494
1
I‐Total
‐
|1
the
to the
closest to
is closest
following is
the following
which of the
random, which
at random,
California is chosen
plumber in California
a plumber
table, ifif a
the table,
on the
Based on
Based
chosen at
experience?
of
years
four
least
at
has
plumber
the
that
probability
probability that the plumber has at least four years experience?
A) 0.10
B) 0.22
C) 0.25
D) 0.46
which
the table,
in the
included in
those included
from those
random from
at random
chosen at
experience is chosen
years of experience
four years
least four
at least
with at
worker with
a worker
IfIf a
table, which
plumber?
a plumber?
person is a
the person
that the
probability that
the probability
to the
closest to
is closest
following is
the following
of the
A) 0.10
B) 0.22
C) 0.25
D) 0.46
_
‐
‐
_
_
_
‐
237
_
A
_
‐
‐
‑
‐-‑
CHAPTER
PROBABILITY
CHAPTER 26 PROBABILITY
Won
Lost
Total
Underdog
Underdog
10
35
45
Favorite
Total
25
5
40
30
75
‐
-
-
35
m
The table
categorized by whether
the
considered the
was considered
team was
whether the team
team, categorized
baseball team,
a baseball
of a
results of
shows the results
above shows
table above
favorite
games in
the games
fraction of the
What fraction
(expected to lose). What
underdog (expected
game or the underdog
in the game
win) in
to win)
(expected to
favorite (expected
team win?
which
the team
did the
underdog did
considered the underdog
was considered
team was
which the team
2
A} -
5
2
B) 7
2
C) 9
2
D) 15
‐ W e e Week
k 11
Week2
W
een
40
mMattresses
m 61
‐
Box springs
‐
Total
‐
-
35
47
82
‐
“
101
‐
Week3e Week4
W
e k Total
s ‑
68
88
-
55
m
77
‑
198
‑
m
348
bedding
weeks at a bedding
four weeks
over four
sold over
units sold
mattress units
and mattress
number of box spring
the number
summarizes the
A store
spring and
manager summarizes
store manager
sold?
units sold?
spring units
fraction of all box spring
what fraction
accounted for what
and 3 accounted
above . Weeks 2 and
table above.
incomplete table
store
store in the incomplete
2
A
A) ) 15‐
4
B
B) ) 15‐
2
C) 5
4
D)
D) 5
238
PANDA
COLLEGE PANDA
THE COLLEGE
THE
Gold
Silver
Bronze
Total
USA
46
29
29
104
China
38
27
23
88
Russia
24
26
32
82
Great Britain
29
17
19
65
Germany
11
19
14
44
Total
148
118
117
383
Country
m
Olympics . If an
Summer Olympics.
London Summer
the 2012 London
at the
awarded at
medal s awarded
distribution of medals
the distribution
shows the
above shows
table above
The table
The
an
the
gives the
which country
table, which
the table,
countries in the
the countries
one of the
from one
random from
at random
chosen at
be chosen
medalist is to be
Olympic medalist
Olympic
country gives
Bronze medalist?
a Bronze
selecting a
probability of selecting
highest probability
highest
medalist?
USA
A) USA
Russia
B) Russia
Britain
Great Britain
C) Great
Germany
D) Germany
Species
Fish Species
Number of Fish
Number
Cartilaginous
Cartilaginous
Bony
Bony
400
300
800
1,200
Philippines
Philippines
New Caledonia
Caledonia
New
produced by
were produced
above were
table above
the table
in the
data in
The data
bony . The
or bony.
cartilaginous or
either cartilaginous
as either
categorized as
be categorized
can be
AU
A
l l fish can
by
species
fish species
each
that
Assuming
Caledonia.
New
and
Philippines
the
in
species
fish
the
studying
biologists
biologists studying the fish species in the Philippines and N e w Caledonia. Assuming that each fish
how
Philippines is
the Philippines
in the
fish in
a cartilaginous fish
catching acartilaginous
of catching
probability of
the probability
being caught, the
chance of beingcaught,
equal chance
an equal
has an
has
is how
Caledonia?
New
one in N
catching one
probability of catching
the probability
than the
greater than
much greater
much
e w Caledonia?
A)
A)
2
15
1
B) 4
3
C) 10
1
D) 3
239
CHAPTER
CHAPTER 26
26 PROBABILITY
PROBABILITY
fires
‐ u g h m g -Lightning-caused
c a u s e d fi m
s
Human-caused fires
East
East Africa
Africa
South
South Africa
Africa
65
65
Total
‑
30
30
‑
Total
135
220
The
incomplete table
wildfires that
that occurred
occurred in ttwo
regions of Africa
Africa in
The incomplete
table above
above summarizes
summarizes the
the number
number of wildfires
w o regions
2014 by cause.
cause. Based
what fraction
fraction of all
human-caused?
Based on the
the table,
table, what
all wildfires
wildfires in East
East Africa
Africa in 2014 were
were human-caused?
11
A ) 24‑
A)
~~
B ) ‑
B)
~!
C ) ‑
C)
11
D ) 15‑
D)
Not defective
Total
Assembly
Assembly Line
L i n eA
A m_ 300
5,700
Assembly
B m_ 500
Assembly Line
LineB
3,500
6,000
4,000
800
9,200
10,000
Defective
Total
A manufacturer
The results
results of each
each assembly
line's quality
quality
manufacturer uses
uses two
t w o assembly
assembly lines
lines to produce
produce refrigerators.
refrigerators. The
assembly line’s
control
are
shown
in
the
table
above.
If
a
refrigerator
from
the
manufacturer
turns
out
to
be
defective,
control are shown the table above. a refrigerator from the manufacturer turns o u t be defective, what
what
is the
probability that
the refrigerator
was produced
Line A?
A?
the probability
that the
refrigerator was
produced by
by Assembly
Assembly Line
A) 5%
5°/o
A)
B) 37.5%
C) 60%
D)
D) 62.5°/o
62.5%
240
THE
THE COLLEGE
COLLEGE PANDA
PANDA
‐
Type of Residence
‑
_
Family members
_
_
_
1
2
3
4 or more
‐Total
‐
Apartment
‐‐
-
-
10
20
8
8
- 46
Duplex
22
Single residence
3
35
12
‐‐
‐
Total
13
45
12
28
18
30
46
138
-n
8
4
n 46
_
m
‑‑
‑
‑
The table
table above
above summarizes
summarizes the
distribution of living
a
The
the distribution
living situations
situations for residences
residences in a
a neighborhood.
neighborhood. If a
duplex in the
neighborhood is to be
be inspected
inspected at
at random,
what is the
probability that
the residence
duplex
random, what
the probability
that the
residence is
the neighborhood
occupied
occupied by no
no more
more than
than 22 family
family members?
members?
2
A) 23
B)
3)
6
23
17
C)
C) 69
17
D )23‑
D)
Number of soil
soil samples
samples
Number
Percent of samples
samples
Percent
with Chemical
with
Chemical A
Area 1
450
8%
8%
Area 2
550
6%
The
who collected
collected soil
areas to
The data
data in the
the table
table above
above were
were produced
produced by
by ecologists
ecologists who
soil samples
samples from
from two
t w o areas
on the
table, what
determine whether
they were
were contaminated
with
Chemical A. Based
determine
whether they
contaminated w
i t h Chemical
Based on
the table,
what proportion
proportion of the
the
soil samples
samples were
were contaminated
contaminated with
with Chemical
Chemical A?
soil
A) 0.067
B) 0.069
C) 0.070
D) 0.072
241
CHAPTER 26
26 PROBABILITY
CHAPTER
PROBABILITY
Test negative
Has v.
virus
Has
..
Does nnot
Does
o t have
virus
have virus
‐_
Total
30
550
580
Test positive
Total
Total
370
50
420
400
600
1,000
1,000
‐_
m
m
The
that is
is designed
when patients
patients are
are
The table
table above
above shows
shows the
the results
results of
of a
a test
test that
designed to
to give
give a
a positive
positive indicator
indicator when
infected
indicator when
when they
o t infected.
to the
the results,
infected with
with a certain
certain virus
virus and
and a negative
negative indicator
they are
are nnot
infected. According
According to
results,
what is the probability
what
probability that
that the test gives
gives the incorrect
incorrect indicator?
indicator?
A)
A) 5°/o
5%
B) 8%
B)
8°/o
C) 10%
D) 12%
D)
Drug
Sugar Pill
Cured
Cured
Not
Not cured
cured
90
25
incomplete table
The incomplete
study in
gave patients
experiencing back
back
table above
above shows
shows the
the results
results of aa study
in which
which doctors
doctors gave
patients experiencing
pain either
pain
either aa drug
as many
many patients
drug
than
from
drug or a
a sugar
sugar pill.
pill. Three
Three times
times as
patients were
were cured
cured from
from the
the drug than from the
the
sugar pill. For every
sugar
by the
the sugar
sugar pill,
pill, 5
patients
were
n
o
t
cured
by
the
According
every 2 patients
patients cured
cured by
5 patients were not cured by the sugar
sugar pill.
pill. According
the results,
results, if a
to the
person will
w i l l be
be cured
cured of
of back
a patient
patient is given
given a sugar
sugar pill, what
what is the
the probability
probability that
that the
the person
back
pain?
pain?
1
A)?I
A) 4
2
B);
B) 7
C) 3
C)13‐0
10
2
D);
D) 5
242
242
‐‐‑
THE
PANDA
COLLEGE PANDA
THE COLLEGE
-Juniors
Seniors
seniors
Total
Gym
equipment
Gym equipment
240
‐
Computers
" 300
160
460
Total
m
540
computers .
equipment or computers.
gym equipment
surplus on nnew
budget surplus
a budget
whether to spend
deciding whether
school is deciding
principal of aa school
The principal
spend a
e w gym
senior
a senior
If a
students . If
senior class students.
and senior
junior and
among junior
preferences among
summarizes the preferences
above summarizes
incomplete table above
The incomplete
How
~- H
equipment is 33.
gym equipment
prefers gym
student prefers
the student
that the
probability that
the probability
random, the
at random,
chosen at
school is chosen
from the school
ow
school?
the school?
are at the
seniors are
many seniors
many
243
243
27
Statistics II
Statistics
list of numbers:
Consider
numbers:
this list
Consider this
5,6,2,2,2,7
5,6, 2, 2, 2, 7
average :
the average:
list is the
the list
The mean
The
m e a n of the
5+ 6 + 2 + 2+ 2+ 7
‐ _ 6
_
4
is 3.
3, 4, 5} is
of {1,
order. For
the list
middle when
the middle
number in the
the number
median is the
The median
The
when the
list is in order.
For example,
example, the
the median
median of
{1, 2,
2,3,4,5}
3.
looks like
which looks
list, which
our
For
For o
u r particular
particular list,
like
2,2,2,5,6,7
2,2, 2, 5, 6, 7
even
an even
there's an
because there’s
That's because
median . That’s
when
there is no
no single
single middle
middle number
number we
we can
can consider
consider the
the median.
ordered, there
when ordered,
numbers,
middle
two
the
of
average
the
When that's
the list
numbers in the
number of numbers
number
list.. When
that’s the
the case,
case, the
the median
median is the average
the t w o middle numbers, 2
and
and 5:
+5
22
T‐+ 5 _ '357
3.5
2
-l..::'.::::J
looking for
we're looking
the median?
determine the
were 100 numbers
list were
the list
Now
numbers long?
long? How
How would
would we
we determine
median? Well, if we’re
for
what if the
Now what
which designates
take half
sense to take
number, it would
the
would make
make sense
half of 100 to get
get 50, which
designates the
the 50th
50th number.
number. But is
is
middle number,
the middle
it's the
numbers. Maybe
number of numbers.
even number
there's an
since there’s
Probably nnot
the median?
number the
50th number
the 50th
the
median? Probably
o t since
an even
Maybe it’s
the average
average
large lists,
with large
See, with
numbers? See,
50th numbers?
and 50th
49th and
the average
numbers . Or is it the
and 51st
the 50th
of the
50th and
Slst numbers.
average of the
the 49th
lists, it's
it’s hard
hard
to tell.
odd or even
there's an odd
whether there’s
median: regardless
the median:
getting the
technique for getting
Here’s
regardless of whether
even number
number of numbers,
numbers,
Here's my technique
have
Since we
up" to 51. Since
"go up”
and “go
divide by 2 to get
numbers, we
So for a list
up." So
always "go
we always
”go up.”
list of 100 numbers,
we divide
get 50 and
we have
For a list of 101
numbers. For
51st numbers.
and 5lst
average the
the average
the median
numbers, 50
whole numbers,
ttwo
w o whole
50 and
and 51, the
median is the
the 50th
50th and
101 numbers,
numbers,
51st
the Slst
median is the
the median
number, 51, the
whole number,
one whole
have one
only have
and "go
get 50.5 and
we divide
”go up"
up” to 51. Since
Since we
we only
divide by 2 to get
we
number.
number .
numbers (the
out on a list of 3 numbers
test it out
works . Just
but it works.
weird but
technique is weird
this technique
Yes, this
Just for reassurance,
reassurance, let's
let's test
(the median
median
numbers).
3rd
and
2nd
the
of
average
the
is
median
(the
numbers
4
of
list
a
and
number)
2nd
the
is obviously
numbers (the median the average the 2nd and 3rd numbers).
obviously the 2nd number) and a list
number
the 2nd
that the
confirms that
which confirms
up" to 2, which
"go up”
then ”go
take half
numbers, we
list of 3 numbers,
For a list
we take
half of 3 to get
get 1.5, and
and then
2nd number
For
numbers
whole
The
3.
to
"
up
"go
then
and
2,
get
to
4
of
half
take
we
numbers,
4
of
list
median. For
For a
numbers,
take half
get and then "go up”
The whole numbers 2
the median.
is the
median.
the
is
numbers
3rd
and
2nd
the
of
average
the average the 2nd and 3rd numbers the median.
and
that the
confirm that
and 3 confirm
particular list,
our particular
the most
up the
The
most often.
often. In our
list, it's
it’s [I].
..
shows up
that shows
number that
the number
mode is the
The mode
244
THE
PANDA
COLLEGE PANDA
THE COLLEGE
number :
smallest number:
the smallest
and the
the list and
biggest number
between the
difference between
the difference
range is the
The
The range
the biggest
number in the
7‐2:‑
7 - 2 =~
how much
words, how
numbers is. In
spread oout
measure of how
a measure
deviation is a
standard deviation
The standard
The
how spread
u t a list of numbers
In other
other words,
much the
the
closer to the
are closer
numbers are
more numbers
when more
lower when
deviation is lower
standard deviation
mean. The standard
the mean.
from the
"deviate" from
numbers ”deviate”
numbers
the mean.
mean.
example, oour
For example,
mean. For
the mean.
from the
away from
out away
spread out
are spread
numbers are
The
The standard
standard deviation
deviation is higher
higher when
when more
more numbers
ur
list
2, 2, 2, 5, 6, 7
following list
the following
deviation than
standard deviation
higher standard
have a higher
would have
would
than the
5,5,5,5,6,7
5, 5, 5, 5, 6, 7
deviation
the standard
that the
turns oout
mean. It turns
the mean.
tightly clustered
more tightly
second list is more
the second
because
clustered around
around the
u t that
standard deviation
because the
we got these
how we
Don't worry
second list is 0.83. Don’t
the second
deviation of the
standard deviation
the standard
and the
list is 2.28 and
of oour
u r list
worry about
about how
these
compare one
know how
Just know
SAT. Just
the SAT.
on the
deviation on
the standard
calculate the
to calculate
asked to
never be
values‐you’ll
be asked
standard deviation
how to
to compare
one
values-you'll never
did.
we just
another's as
with another's
deviation with
list's standard
list’s
standard deviation
aswe
just did.
EXAMPLE1:
EUMPLB 1:
. . .'
Sports
Playing Sports
Spent Playing
Daily Hours
Hours Spent
Daily
i
1---130
tl)
o
...
fz
20 --10 -0
I•
1
2
Hours
3
'-·.
'L
at aa
students at
summarizes the
above summarizes
The histogµun
histogram above
the daily
daily number
number of hours
hours spent playing
playing sports
sports for 80 students
school.
school.
80 students?
sports for the 80students?
spent playing
daily number
the mean
PART 1: What
What is
is the
mean daily
number of hours
hours spent
playing sports
PART
students?
sports forthe'80
playing sports
spent playing
daily number
the median
PART 2: What
median daily
number of hours
hours spent
for the 80students?
What is the
number of
the number
that by the
divide that
Then divide
student. Then
every student.
number of hours
total number
up the total
Sum up
hours for every
Solution: Sum
Part 1 Solution:
students..
students
Total
hours
(0x5)+(1x35)+(2x15)+(3><25)
_ 140 =
_ I1.75 I
x 5) + (1 x 35) + (2 x 15) + (3 x 25) =
= (0
Total hours
80
80
students ‐
Number
_
Number of students
histogram
(the histogram
middle (the
the middle
are the ttwo
students are
41st students
and 4lst
the 40th
students, the
80 students,
group of 80
Part 2 Solution:
Solution: In a group
40th and
w o in the
hours playing
spend 0 hours
students spend
have to). The first 5 students
don't have
their hours
already
students by their
hours so
so we don’t
playing
the students
orders the
already orders
40th student
so the 40th
student, so
includes the 40th
group includes
sports
next 35 students
students spend
spend 1 hour.
hour. This group
40th student,
student
each day. The next
sports each
41st
the
so
student,
41st
the
includes
group
Now
hours.
2
spends
1
hour.
The
next
15
students
spend
hours.
N
o
w
this
group
includes
the
4lst
student,
so
the
4lst
spend
students
15
next
spends hour.
student
spends
2
hours.
Taking
the
average,
average,
Taking
hours.
student spends
student _ 1 + 2 _ l1ci
Daily
spent by 40th
40th student
student + Daily
Daily hours
l s t student
spent by 441st
hours spent
hours spent
Daily hours
-~
2 2
-= 2
2
245
CHAPTER
STATISTICS I
CHAPTER 27 STATISTICS
"EXAMPL'E2:
EXAMPLEZ:
/
O
O
O
O
~-
•1
., .
':~f·•.,1
·, ,: 11-1
I
,,'
~
'••J, :/
I
+'!
0
0
0
O
o
o
o
o
0o
5
4 5
2 33 4
11 2
Flights'I‘akeninaYear
Taken ih a Year
Flig,hts
6
O
\ I
.!
O“
Q
0
0
Q
o
I\
,
•6
·.•
I•
I
.,
•
,
student
students. If
taken in a year by 19 college students.
The dot
plot above
the number
of flights
flights taken
If the
the student
number-of
sumtnarizes the
above summarizes
dot P,lot
describes the
correctly describes
following correctly
the following
which of the
datfl, which
took 6 Bights
flights in
in a
is remov-ed
removed from
from the data,
year is
a year
who -took
changes
to the statistical measures of the data?
data?
chc:Utges
. ·
d~as~s
·· I.I Tue
The tneatt
mean1decreases.
-\ t
/"'
decreases.
n.
I I . The median
median decreases.
d~creas~ ~
Therartge
III. The
range decreases.
1-¥'.J.
. : , >''1
114 i~ I'
~
,,.m.
1,»leme
y
B)Iandllon1y
B.)I and.ll.ol'.d
.r •~
•
qr.andIDQnly
C)IandIIIon1y
, ll,andW
D)I
D)I,II,andIlI
I ,I
where
outside where
is far outside
that is
point that
data point
extreme data
an extreme
an outlier,
year is
flights in a year
took 6 flights
who took
The
is called
called an
outlier, an
student who
The student
up . It
(mean)
average
the
brings the average (mean) up.
data, itit brings
rest of the data,
than the
this outlier
Because this
data lies.
most of the
lies. Because
outlier is greater
greater than
the rest
the data
most
maximum (6).
the maximum
(0) and
mi:nimwn (0)
the minimum
since there
range since
the range
increases the
also increases
also
there is a larger
larger gap
gap between
between the
and the
The median,
decreases. The
the range
and the
decreases and
mean decreases
the mean
outlier is removed,
this outlier
When this
When
removed, the
range decreases.
median, however,
however, is unaffected
unaffected..
is
and the
students, and
19 students,
are 19
there are
outlier is removed,
the outlier
Before the
calculate it. Before
let's calculate
this, let’s
confirm this,
To confirm
removed, there
the median
median is
and
students,
18
are
there
removed,
is
outlier
the
After
flight.
one
took
who took one flight. After the outlier removed, there are 18 students, and
student, who
the 10th
represented
represented by the
10th student,
does
median of 1 does
the median
So the
flight. 50
one flight.
whom took one
and 10th
the
represented by the
the 9th
9th and
10th students,
students, both
both of whom
median is represented
the median
median. Answer
but
the mean
typically affect the
outliers typically
n
o t change.
n d in fact, outliers
mean b
u t not
n o t the
the median.
A n s w e r [@]
(C) .
And
change. A
not
w.eighing 230
panda, weighing
Anqther panda,
pounds. Another
200· pounds.
ts 200
a group
avetage weight
EXAMPLE 3:
3: The average
weight of a
group of p'illt~~
pandas is
~PLE
matty pandas
How
..
poun.ds H o w many
group to 205 pounds.
entire ..group
average wej,ght
grqup., :rais-mg
oins:the
pounds,jjoins
the group,
raising the
the average
weight of the
the entire
pandas
pQ~ds
were
group? .
,
Qti~a.l \~oup'?
fl),eoriginal
ht the
wei;e·in
with
less to do w
have less
questions have
These questions
averages . These
problem that
a word
you will
while, you
Once
will get a
word problem
that involves
involves averages.
ith
Once in a while,
cover
to
decided
chapter, we decided
this chapter,
averages in this
with algebra,
more
and m
statistics and
o r e to do with
algebra, but
but because
because we cover
cover averages
cover
statistics
as well.
here as
problems here
these
well.
word problems
types of word
these types
always find the
totals. You can always
sums or totals.
think in terms of sums
questions on the SAT, think
average questions
When
i t h average
with
dealing w
When dealing
average with
s u m by multiplying
multiplying the average
with the number
number of subjects.
subjects.
sum
When
200x . When
then 200x.
group is then
original group
the original
weight of the
total weight
be x.
Let the
pandas in the
the original
original group
group be
r. The
The total
number of pandas
the number
).
1
+
weight is 205(x + 1).
total weight
the total
and the
+ 1 and
number of pandas
another
the group,
group, the
the number
pandas is x +
panda joins the
another panda
pounds ,
Since
panda weighs
weighs 230 pounds,
that panda
Since that
200x + 230 =
: 205(x + 11))
200x +
+ 230 = 205x ‐l‐
+ 205
= -‐ 225
5
-‐ 55xx =
: 5
S
x=
group .
There
original group.
the original
panda s in the
were [fil pandas
There were
246
246
COLLEGE PANDA
THE COLLEGE
PANDA
THE
EXAMPLE4:
.,, ,·
Neighborhood B
Neighborhood A
r,._•
...
j......30
!
cl 20
t
l
~
I "'
10
0
'0‘123456
1 2 3 4 5 6
awned
of cars owned
Number
Numberofcars
Nw;nber
Number of cam
cars owned
owned
and B,
neighborhoods, A
that residents
charts above
The bar
bar charts
above summarize
summarize the
the number
number of cars that
residents from ttwo
w o neighborhoods,
A and
B,
by
wned
o
cars
of
m1mbei
the
of
deviation
standard
the
correctlycompares
the following
own.
Which of the
following correctly
compares the standard deviation the number of cars ownedb
own. Which
•
·
neighborhoods?
the neighborhoods?
tesidents
residents in each of the
.
larger.
is
A
eighborhood A is larger.
ca:rsowned
deviation of the
standard deviation
fhe standard
A) The
the number
number of cars
owned by residents
residents in
in Neighborhood
is larger.
Neighborhood
in
residents
by
oWhed
cars
of
number
the
o.fi
deviation' of
B) The
The standard
standard deviation
number cars owned residents Neighborhood"BBis
larger.
B)
and
·residents .in
owned by residents
deviation of the number
standard deviation
C) The standard
number of cars owned
in Neighborhood
Neighborhood A
A and
same.
is the same.
Neighborhood B
Bis
.
Neighborhood
determined from
cannot be
relationship cannot
be determined
from the information
information given.
given.
D) The relationship
which,
mean, which,
the mean,
out from
much more
are much
Neighborhood B are
data for Neighborhood
the data
Most
Most of the
are at
at the
the ends
ends and
and are
more spread
spread out
from the
A,
Neighborhood
for
The data
cars. The
be 3 cars.
estimate to be
can estimate
symmetrical, we
bar graph
because the
graph is symmetrical,
we can
data
Neighborhood A, on
on the
the
the bar
because
for
deviation
standard
the
,
is. Therefore
mean
the m
where the
are more
other hand,
other
hand, are
more clustered
clustered towards
towards the
the low
low end,
end, where
e a n is.
Therefore, the standard deviation for
larger. Answer
Neighborhood
Answer ~-(B) .
B is larger.
Neighborhood Bis
Boxplots
Boxplots
are just
and dotplots,
histograms and
SAT. Like histograms
the SAT.
on the
shows up
question shows
boxplot question
then , aa boxplot
and then,
Every
Every nnow
o w and
up on
dotplots, boxplots
boxplots are
just
one:
construct
to
need
you
that
metrics
statistical
5
are
There
.
data
numerical
visualizing
of
another
another way
way
visualizing numerical data. There are 5 statistical metrics that you need to construct one: the
the
maximum .
the maximum.
and the
the first quartile
minimum,
m
i n i m u m , the
quartile,, the
the median,
median, the
the third
third quartile,
quartile, and
weights of 30 tortoises:
the weights
that summarizes
example of a boxplot
Here’s
boxplot that
summarizes the
tortoises:
an example
's an
Here
Ql
Q1
Min
Min
t
‐
‐
(
r
+
‐+‐‐
10
10
20
20
30
30
Q3
Q3
Median
Median
:
:
q
l
+
‐
Max
‐
+
50
40
40
50
60
Weight (in pounds)
pounds)
Weight
i
1
70
70
l
80
line segment
the line
and the
quartiles, respectively,
third quartiles,
and third
The left and
and right
right sides
sides of the "box"
”box” represent
represent the
the first and
respectively, and
segment
The
the
to the
box to
the left and
drawn from
are drawn
median. Line segments
the median.
inside the
indicates the
segments are
from the
and right
right sides
sides of
of the
the box
box indicates
the box
inside
whiskers.
called whiskers.
These segments
minimum and
and the
maximum.. These
segments are
are sometimes
sometimes called
the maximum
minimum
quartile
pounds, the
tortoises is 15 pounds,
the 30 tortoises
among the
From the
the boxplot,
boxplot, we can
can see
see that
that the
the minimum
minimum weight
weight among
the first quartile
From
pounds
70
is
maximum
the
and
pounds, and the maximum is 70 pounds..
60 pounds,
the third
pounds, the
40 pounds,
median is 40
is 30
30 pounds,
third quartile
quartile is
is 60
the median
pounds, the
is
247
CHAPTER 27 STATISTICS
CHAPTER
STATISTICS I
N
o w before
we get
let’s go
go over
over what
what the
and third
Now
before we
get too
too far ahead
ahead of
of ourselves,
ourselves, let's
the first
first quartile
quartile and
third quartile
quartile mean.
mean.
The
first quartile,
also known
known as
as the
the lower
lower quartile
quartile or
or 25th
25th percentile,
The first
quartile, also
percentile, is
is the
the value
value for
for which
which 25%
25% of
of the
the data
data is
is
less
than.
If
your
score
on
an
exam
is
equal
to
the
first
quartile,
then
you
scored
better
than
25%
of
the
people
less than. If your score on an exam is equal to the first quartile, then you scored better than 25% of the people
who
who took
took the exam.
exam .
The
known as
upper quartile
quartile or
or 75th
75th percentile,
value for
which 75%
data is
is
The third
third quartile,
quartile, also
also known
as the
the upper
percentile, is
is the
the value
for which
75% of
of the
the data
less
If your
your score
score on
on an
equal to
to the
the third
quartile,
then
you
scored
better
than
75%
of
the
people
less than.
than. If
an exam
exam is
is equal
third quartile, then you scored better than 75% of the people
who took
took the
who
the exam
exam..
To calculate
calculate the
these steps
steps::
the quartiles,
quartiles, follow these
1. Make
Make sure
sure the
the data
data set
set is ordered.
ordered .
2. Find
Find where
where the
the median
median is.
3. Use
Use the median
median to split
data set
into two
t w o halves
halves.. Do nnot
o t include
50 when
when
split the
the data
set into
include the
the median
median in
in either
either half.
half . So
the data
data set
contains 7 numbers,
the
set contains
numbers, the
halves are
are the
the first 3 numbers
and the last
last 3 numbers
since
the two
two halves
numbers and
numbers since the
the
4th element
4th
be excluded.
excluded. When
the
data
set
contains
8
numbers, the
the two
two
element is the
the median
median and
and needs
needs to be
When
data
contains 8 numbers,
halves are
are the
halves
the first 4 numbers
numbers and
last 4 numbers.
numbers. The median
”excluded” since
since it's
it's the
and the last
median is already
already "excluded"
the
average of the
average
the 4th
4th and
numbers.
and 5th
5th numbers.
The first quartile
4. The
quartile is the
median of the
lower half
half of the
is the
the median
median of
of the
upper
the median
the lower
the data.
data . The third
third quartile
quartile is
the upper
half
the data.
data.
half of the
Note that
that there
there are
are actually
Note
actually several
mathematicians use
use to calculate
quartiles, and
the resulting
several ways
ways mathematicians
calculate the
the quartiles,
and the
resulting values
values
can differ
differ depending
depending on
on the
can
the method.
method. Fortunately,
Fortunately, this
this isn't
isn’t something
need to worry
something you
you need
worry about.
about. For
For the
the
purposes of the
the SAT,just
the method
purposes
just use
use the
method described
described above
above and
and you’ll
you'll be
be fine.
illustrate, let’s
let's do
an example.
example. Let's
Let's construct
To illustrate,
do an
from the following
prices for 15
15 different
different
construct a boxplot
boxplot from
following set of prices
textbooks:
textbooks:
{24,28,30,30,30,72,75,82,88,90
{ 24,28, 30, 30, 30, 72, 75, 82, 88, 90,
100,100, 100, 100, 130}
,100,100,100,100,130}
o
The m
i n i m u m iiss 24.
• The
minimum
o
15+
2 = 7.5 ‐>
number: 82.
• The
The median
median is the
the 15
-:-2
-+ 8th
8th number:
• Because
Because the median
median is the 8th number,
0
number, we can
can use
use itit to split
into ttwo
w o halves:
split the data
data set into
halves: the
the first 7
numbers
and the
numbers and
the last
last 7 numbers.
numbers. We do not
not include
include the
the median
half..
median in either
either half
• The first quartile
quartile is the
the median
median of the
0
the first 77 numbers,
+ 2 z= 3.5 ‐>
numbers, so
so it’s
it's the 7
7-:-+ 4th
4th number:
number: 30.
• The
The third
third quartile
o
numbers, so
so it’s
+2=
= 3.5 -+
‐> 4th
4th number
number from
from the
the
quartile is the
the median
median of the
the last
last 7 numbers,
it's the 77 -:-2
median or the
the 8 +
median
+ 4 = 12th
12th number
number in the
the overall
overall set: 100.
maximum is
is 130.
•o The maximum
Using these
these values,
values, we can
Using
can n
o w construct
boxplot:
now
construct the
the boxplot:
H _ _ _ J _ l ‐ H
t
‐
t
‐
i
‐
t
‐
H
‐
t
‐
t
‐
t
‐
t
‐
f
‐
t
‐
F
‐
‘
l
200 3300 440
2
0 550
0 660
0 70
0 11110
0 1 2120
0 1 130
3 0 1140
40
70 8 0809 0901 0100
Price
Price
248
THE COLLEGE
PANDA
THE
COLLEGE PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 338.
A calculator is allowed
allowed on the following
following
questions.
questions.
The
average height
students in one
The average
height of 14
14students
one class
class is
is
63
63 inches
inches.. The
The average
average height
height of 21
21 students
students in
another
classes are
another clas
classs is 68. If
If the
the two
t w o classes
are
combined , what
the average
inches,
combined,
what is
is the
average height,
height, in inches,
of the
the students
students in the
the combined
combined class?
class?
.....
0
....
QJ
"E
2
:l
z
A) 64.5
B)) 665
5
0
C)) 666
6
J, "'-~
~
D)) 666.5
6.5
~
~
{'
?:,~
lo' ......
~ ...,_lo'
'I,~ "v'd"
Books read
The histogram
histogram above
above shows
The
shows the
the number
number of
books
read last
last year
20 editors
a publishing
publishing
books read
year by
by 20
editors at
at a
company. Which
Which of the
the following
following could
be the
the
company.
could be
median number
number of books
books read
read by the
20 editors?
median
the 20
editors?
Kristie
taken five tests
Kristie has
has taken
tests in science
science class
class.. The
The
average of all
test scores
average
all five of Kristie's
Kristie’s test
scores is 94.
The
The average
average of her
her last
last three
three test
test scores
scores is 92.
What
s?
What is the
the average
average of her
her first
first ttwo
w o test
test score
scores?
A)) 110
A
0
B)) 1122
B
A ) 995
5
A)
C
6
C)) 116
B) 96
B)
D ) 223
3
D)
C)
C ) 9977
D ) 9988
D)
Final Exam
Exam Scores
Scores
Final
(out
(out of 100 points)
point s)
A food
food company
company hires
hires an
an independent
independent research
research
agency
agency to determine
determine its
its product's
product’s shelf
shelf life,
life, the
the
length
length of time
time it may
may be
be stored
stored before
before it expires.
expires.
Using
random sample
units of the
Using aa random
sample of 40
40 units
the
product, the
the research
product,
research agency
agency finds
finds that
that the
the
product's
s. Which
product’s shelf
shelf life
life has
has aa range
range of 33 day
days.
Which
of the
st be
the following
following mu
must
be true
true about
about the
the units
units in
the sample?
sample?
the
l~
‐ i : l : l ‐ ‐ l
t
‐
70
t
‐
75
‐
‐
+
80
‐
t
‐
85
‐
l
‐
90
‐
‐
l
95
‐
‐
l
100
The box
box plot
plot above
above summarizes
summarize s the
the final
final exam
exam
The
scores
studentss in a
math class.
Based on
on
scores of 26
26 student
a math
class . Based
the
following best
the box
box plot,
plot, which
which of the
the following
best
estimates the
the number
the
estimates
number of points
points by which
which the
median
score of the
students exceeds
exceeds the
median score
the 26
26 students
the
lowest individual
individual score?
lowest
score?
A) All
All the
the units
units expired
expired within
within 33 days
days..
B)
The unit
unit with
with the
the longest
longest shelf
shelf life
life took
took 33
B) The
days
unit with
the
days longer
longer to expire
expire than
than the
the unit
with the
shortest shelf
shelf life
life..
shortest
A) 3
The mean
mean shelf
shelf life
life of the
the units
units is 3 more
more
C) The
than the
the median.
median.
than
B) 5
C
C)) 8
The median
shelf life
units is 3 more
more
D) The
median shelf
life of the
the units
than the
mean.
than
the mean.
D ) 11
D)
11
249
CHAPTER
STATISTICS I
CHAPTER 27 STATISTICS
Locks are
are sections
sections of canals
canals in which
which the
the water
water
level
to raise
level can
can be
be mechanically
mechanically changed
changed to
raise and
and
lower
shows the
number
lower boats.
boats. The
The table
table below
below shows
the number
of locks
locks for 10 canals
canals in France.
France .
Miss World
World Titleholders
Miss
Titleholders
••• •• •
••• ••• ••• ••• •• ••• •
0
0
0
0
0
O
O
0
0
O
0
0
0
O
O
0
0
Name
I # Locks I
Q
0
0
0
O
O
0
Aisne
27
18
18
19
19
22
20 21
21 22
20
Age
(years)
Age (years)
23
23
24
24
Alsace
25
Rhone
5
Centre
30
Garonne
Lalinde
23
The dotplot
dotplot above
above shows
the distribution
distribution of ages
The
shows the
ages
24 winners
winners of the
the Miss
Miss World
World beauty
beauty pageant
pageant
for 24
at the
the time
time they
they were
were crowned.
crowned. Based
at
Based on
on the
the
data, which
which of the
the following
following is
to the
data,
is closest
closest to
the
average (arithmetic
age of the
the winning
winning
average
(arithmetic mean)
mean) age
Miss
World pageant
pageant contestant?
contestant?
Miss World
A)) 119
A
9
Midi
27
32
Oise
27
Vosges
93
29
Sambre
B) 20
20
B)
Removing which
which of the
the following
following ttwo
canals
Removing
w o canals
from the
the data
data would
would result
result in the
the greatest
from
greatest
decrease in the
the standard
decrease
standard deviation
deviation of the
the
number of locks
locks in each
number
each canal?
canal?
C)) 2211
C
D)) 2222
D
Temperature (°F)
Frequency
60
3
61
4
63
4
67
10
70
7
Aisne and
and Lalinde
Lalinde
A) Aisne
8) Alsace
Alsace and
and Garonne
Garonne
B)
C) Centre
Centre and
and M
Midi
idi
Rhone and
D) Rhone
and Vosges
shoe store
store surveyed
surveyed a
a random
random sample
A shoe
sample of 50
50
customers to better
better estimate
estimate which
which shoe
shoe sizes
customers
sizes
should be
be kept
kept in stock.
stock. The
The store
store found
found that
that the
the
should
median shoe
shoe size
size of the
the customers
customers in the
the sample
sample
median
inches. Which
Which of the
the following
following statements
statements
is 10 inches.
must
be
must
true?
true?
The
above gives
the distribution
The table
table above
gives the
distribution of low
low
temperatures for a
a city over
over 28
28 days.
days . What
the
temperatures
What is the
median
low temperature,
temperature, in degrees
degrees Fahrenheit
Fahrenheit
median low
(°F), of the
the city for these
these 28
28 days?
days?
(°F),
The sum
sum of all the
the shoe
shoe sizes
the sample
sample
A) The
sizes in the
inches .
is 500 inches.
average of the
smallest shoe
size and
and
B) The average
the smallest
shoe size
the largest
largest shoe
the sample
sample is 10
the
shoe size
size in the
10
inches.
inches.
C) The
The difference
difference between
between the
the smallest
smallest shoe
shoe
C
size and
and the
the largest
largest shoe
shoe size
size in the
the sample
size
sample
inches.
is 10
10inches.
D) At
At least
least half
half of the
the customers
customers in the
the sample
sample
D
have shoe
sizes greater
greater than
than or equal
equal to 10
have
shoe sizes
10
inches.
inches.
250
__
The
The tables
tables below
below give
give the
the distribution
distribution of travel
travel
times
times between
between ttwo
w o towns
towns for Bus
Bus A and
and Bus
Bus B
40 days .
over
over the
the same
same 40days.
Bus
Bus A
Travel
Travel time
time (minutes)
(minutes)
_‑
44
45
47
48
THE
PANDA
THE COLLEGE
COLLEGE PANDA
10 ~~---_-_-_-_-_-_-__-_-_-__:;---
---~
9
-Company
A 1---------i
8 _ i::::::::::J
Company B ,_______
A28
_.
7
“5 7
an
Frequency
6
5
-g
... 5
10
10
.o
0
Q)
5
4
53
1
zZ 2
15
10
2
1
0
45
Bus B
Travel
Travel time
time (minutes)
(minutes)
Frequency
25
5
30
35
40
5
47
48
46
Weight
Weight (in
(in pounds)
pounds)
49
The bar
bar chart
above shows
the distribution
The
chart above
shows the
distribution of
weights (to
(to the
the nearest
pound) for
for 19 kayaks
weights
nearest pound)
kayaks
made
and 19 kayaks
kayaks made
made by
made by
by Company
Company A and
by
Company B.
B. Which
of the
the following
Company
Which of
following correctly
correctly
compares
the median
median weight
the kayaks
compares the
weight of the
kayaks made
made
by each
each company?
company?
10
15
10
Which
Which of the
the following
following statements
statements is true
true about
about
the
the data
data shown
shown for these
these 40 days?
days?
A) The
the kayaks
The median
median weight
weight of the
kayaks made
made by
by
Company A is smaller.
smaller.
Company
A) The
The standard
standard deviation
deviation of travel
travel times
times for
for
Bus
Bus A is smaller.
smaller.
B) The
The median
median weight
weight of the
the kayaks
kayaks made
made by
by
Company B is smaller.
smaller.
Company
B)
The standard
standard deviation
deviation of travel
travel times
times for
for
B) The
Bus Bis
B is smaller
smaller..
median weight
kayaks is
C) The
The median
weight of the
the kayaks
is the
the
same
companies .
same for both
both companies.
C)
C) The
The standard
standard deviation
deviation of travel
travel times
times is
the
the same
same for Bus
Bus A
A and
and Bus
Bus B.
D)
D) The relationship cannot be determined
from the information given.
D)
travel times
The standard
standard deviation
deviation of travel
times for
D) The
Bus
and Bus
Bus A and
Bus B cannot
cannot be
be compared
compared with
with
the
provided .
the data
data provided.
Quiz
Score
1
87
2
75
3
90
4
83
5
98
6
87
7
91
The
above shows
shows the
the scores
scores for
Jay's first
The table
table above
for Jay’s
first
seven
quizzes . Which
the following
following are
are
seven math
math quizzes.
Which of
of the
true
about his
his scores?
true about
scores?
1, The
The mode
mode is greater
greater than
than the
the median.
I.
median .
n. The
greater than
than the
the mean.
mean .
II.
The median
median is greater
III. The
range
is
greater
than
20.
The range greater than
A)
B)
C)
D)
251
llII only
only
IIII
I I only
only
II and
and III
III
JI
I, H,
II
11,and
and IIII
STATISTICS I
CHAPTER 27 STATISTICS
CHAPTER
5
Vl
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8
7
Integers
Integers
1
0
910
1
The
frequency
the frequency
shows the
above shows
graph above
The graph
generated
randomly generated
list of randomly
distribution
distribution of aa list
the
Which of the
integers
and 10. Which
between 5 and
integers between
range
following
the mean
the range
and the
mean and
gives the
correctly gives
following correctly
of the
integers?
list of integers?
the list
A)
B)
C)
D)
4
u
......
3 1----------
2
4
3
5
shown
films
of
Number
The
number of films
the number
shows the
above shows
chart above
bar chart
The bar
shown
classes
year for 19 classes
past year
the past
over the
class over
shown in class
in School
Which of
School B. Which
15 classes in School
and 15classes
School A and
and
the
mean and
the mean
compares the
correctly compares
following correctly
the following
class for
each class
shown in each
median
number of films shown
median number
schools?
the ttwo
the
w o schools?
Mean
=4
Range =
= 7.6, Range
Mean =
Range
7.6,
=
Mean :
Mean
Range := 5
median number
and median
A) The
e a n and
number of films
mean
The m
both greater
are both
class are
each class
shown
greater in
shown in each
School
A.
School A.
Mean
= 8.2, Range
Range == 4
Mean =
Range =
Mean
: 5
Mean = 8.2, Range
number of films
median number
and median
mean and
The mean
B) The
shown
greater in
both greater
are both
class are
each class
shown in each
School
School B.
each
C) The
shown in each
films shown
number of films
mean number
The mean
median
the median
class
School A, but
but the
greater in School
class is greater
is the
schools.
both schools.
same in both
the same
Meals
Calories in Meals
Calories
500
500
520
550
550
550
550
600
600
900
mean number
D) The
number of films shown
shown in each
each
The mean
the median
but the
School B, but
greater in School
class
median
class is greater
schools.
both schools.
same in both
the same
is the
calories in
number of calories
the number
lists the
above lists
The
table above
The table
900-calorie
a 900-calorie
If a
meals. If
last 10
's last
each
Mary's
10 meals.
each of Mary
the values
added to the
today is added
had today
she had
meal
values
that she
meal that
statistical
following statistical
the following
listed,
which of the
listed, which
measures
w i l l n_o_t
change?
not change?
data will
the data
measures of the
1,
Median
I. Median
Mode
Mode
III. Range
Range
1],
II.
A
A))
B
B))
C)
C)
D
D))
lIaand
n d 11
only
II only
[I aand
n d 111
only
III only
only
and IIII
11and
II
I I only
and IIII
II,, II,
I I , and
II
252
THE
COLLEGE PANDA
PANDA
THE COLLEGE
••
•• ••• ••• ••• •
‐‑
•
2 1 222
2 23
23 2
2 5 26
26 2
2 8 29
2 9 30
30
21
244 25
277 28
Gas mileage
mileage (miles
(miles per
per gallon)
gallon)
Gas
The dotplot
dotplot above
above gives
gives the
the gas
gas mileage
(in
The
mileage (in
miles per
per gallon)
gallon) of 15
15different
cars. If the
the dot
dot
miles
different cars.
representing the
the car
car with
with the
greatest gas
gas
representing
the greatest
mileage is removed
from the dotplot,
dot-plot, what
what will
will
mileage
removed from
happen to the
the mean,
mean, median,
median, and
and standard
standard
happen
deviation
deviation of the
the new
n e w data
data set?
Number of
lectures
lectures
Number of
professors
professors
12
15
21
25
28
32
40
15
12
6
20
17
15
5
“‑
The
summarizess the
distribution of
The table
table above
above summarize
the distribution
the
number of lectures
the number
lectures each
each of the
the 90
90 professors
professors
at
last year.
year. Which
Which of the
at acollege
a college gave
gave last
the
following
box plots
data
following box
plots correctly
correctly represents
represents the
the data
shown in the
the table?
shown
table?
A) Only
Only the
the mean
mean will
will decrease
decrease..
B) Only
and standard
Only the
the mean
mean and
standard deviation
deviation will
will
decrease
decrease..
A)
A)
H'-_---'-_
the mean
and median
C) Only
Only the
mean and
median will
will decrease
decrease..
D) The
mean, median,
and standard
standard deviation
The mean,
median, and
deviation
will decrease
will
decrease..
_:------1
l ‐ ‐ 0 ‐ ‐ f ‐ 0 ‐ f ‐ + ‐ ‐ f ‐ l * l ‐ ‐ l
0 1 15
5 20
2 02
3 5 40
4 04
5 0
5 1 10
25 5 3
30 0 35
45 5 50
Number
lectures
Number of lectures
B)
B)
l ‐ E I Z l ‐ l
Snowfall
Snowfall (in
( i n inches)
inches)
mmmmmm
45
48
49
55
57
60
60
57
61
50
57
52
54
58
65
59
i
i
‐
f
‐
O
‐
l
‐
l
‐
M
‐
l
‐
l
5 1 10
0 1 15
5 20
2 0 25
2 5 30
3 03
4 04
5 0
35 5 40
45 5 50
Number
lectures
Number of lectures
mmnmmm
m m m m
61
‐
90
C)
The
table above
The table
above lists
lists the
the amounts
amounts of snowfall,
snowfall, to
the
nearest inch,
experienced by 18
the nearest
inch, experienced
18 different
different
cities
the past
past year
cities in the
year.. The
The outlier measurement
measurement
of 90
the mean,
90 inches
inches is an
an error
error.. Of the
mean, median,
median,
and range
and
range of the
the values
values listed,
listed, which
which will
will change
change
the most
the
most if
if the
the 90-inch
90-inch measurement
measurement is replaced
replaced
by the
correct measurement
measurement of 20 inches?
the correct
inches?
F E D ‐ i
H
t
‐
t
‐
t
‐
t
‐
t
‐
t
‐
t
‐
+
‐
+
‐
~
I
5 1 10
0 1 15
5 2 20
0 2 25
5 3 30
0 3 35
5 440
0 445
5 550
0
Number of lectures
lectures
Number
D)
D)
A) Mean
Mean
Median
B) Median
r
5
Range
C) Range
None of them
them will
will change.
change.
D) None
253
o
H
+
o
+‐
i‐
r
+‐
o ‐4
10 15
15 20 25
25 30
30 35
35 40
45 50
50
40 45
10
Number
lectures
Number of lectures
e
•
Statisti
Statistics
II
cs II
The goal
goal of statistics
statistics is to be
be able
able to make
make predictions
predictions and
information..
and estimations
estimations based
based on
on limited
limited time
time and
and information
might want
the mean
mean weight
of all
all female
female raccoons
raccoons in
in the
United
For example,
example , aa statistician
statistician might
want to estimate
estimate the
weight of
the United
States.
to survey
survey the
the entire
raccoon population
population.. In
In fact,
fact, by
by the
States . The problem
problem is that
that it’s
it's impossible
impossible to
entire female raccoon
the
time
be accomplished,
accomplished, not
n o t only
only would
would the
the data
data be
u t of
would be
e w females
in
time that
that could
could be
be oout
of date
date but
but there
there would
be nnew
female s in
the population.
population. Instead,
Instead, aa statistician
statistician takes
random sample
sample of female
make an
an estimation
of
the
takes aa random
female raccoons
raccoons to
to make
estimation of
what
the actual
actual mean
mean might
might be
be.. In other
words, the
the sample
sample mean
the population
population mean
mean..
what the
other words,
mean is
is used
used to
to estimate
estimate the
Using a
sample to predict
predict something
o m m o n theme
theme in
in stati
statistics
and in
in SAT
SAT
Using
a sample
something about
about the entire
entire population
population is
is aa ccommon
stics and
question s.
questions.
EXAMPLE
EXAMPLE 1: A pet
pet food store
store chose
chose 1,000 customers
customers at _random
random and
many pets
pets
and asked
asked each
each customer
customer how
how many
he
he or she
she has.
has. The results
results are
shown in the
the table
table below.
are shown
I f
t •
Number of pets
Number of customers
1
600
200
100
100
2
3
4ormore
There are
There
are a total
total of 18,000customers
customers in the
the store's
store’s database.
database. Based
Based on the
the survey
what is the
the expected
survey data,
data, what
expected
total number
number of customers
customers who
who own
o w n 2 pets?
pets?
total
Using
sample data,
total number
number who
who o
w n 2 pets
be
Using the
the sample
data, we can
can estimate
estimate the
the total
own
pets to be
200 _ ~
18, 000 X
x ] ,()()Q = ~
3, 600
18,000
1,000 _
254
THE
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120
100
110
130
Heart
rate (beatsperi‘rdnute)
(beats per :i;ninute)
Heartrate
..
. sca~tot
heart-rate
;t.6diffeten;
W W above
W shows
W the
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W M bet;ween
M W
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O oxyg~
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E Mpoiµts
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fineotbest fit is a1s6showi;i.
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xmmu
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at a."be.,rrr!lfeof11Pbeats
PARTl:
_ d on the ,line of best-nt, -;what.isKyle's predicted o~gen ~e
M 1 : Medonthelumetbestfitwhathyle’spredmdoxygmuptdteataheertrateofllobeats
permin,ute?
perndnute?
by
PAR'l'
2: ''what
lS the oxygen1,1ptake,in
lliers per Jll.UlUte,ofthemeasurementiepresentedbythedaw
of the me-asurementreP.~~ted
the data
M12:
Whatistheoxygenuptake,
mittetsperminute,
,oinf
is farthest,fromthe line of best fit?
pomtfimtisfarfiiestfiomthelineofbestfit?
: tnat
.
;
.,,
Part
Solution: Using
that at
at a heart
rate of 110 beats
beats per
per minute
(along the
the
Part 1 Solution:
Using the
the line
line of best
best fit, we
we can
can see
see that
heart rate
minute (along
x-axis),
the oxygen
x-axis), the
oxygen uptake
uptake is ~
liters
liters per
per minute
minute..
Using
a prediction
dangerous, especially
especially when
Using the
the line
line of best
best fit to
to make
make a
prediction can
can be
be dangerous,
when
we are
are making
making aa prediction
prediction outside
outside the
the scope
scope of oour
u r data
oxygen uptake
uptake at
heart rate
rate
•0 we
data set (predicting
(predicting the
the oxygen
at aa heart
le-you'd probably
dead).
of 250 beats
beats per
per minute,
minute, for
for examp
example‐you’d
probably be
be dead).
there are
are outliers
outliers that
may hea
heavily
influence the
the line
Part 2).
•0 there
that may
vily influence
line of best
best fit (see
(see Part
•o the
curve rather
than aa linear
one . In this
this case,
case, a
a
the data
data is
is better
better modeled
modeled by
by aa quadratic
quadratic or
or exponential
exponential curve
rather than
linear one.
linear
like compound
compound interest
linear at
linear model
model looks
looks to be
be the
the right
right one,
one, but
but something
something like
interest may
may look
look linear
at first even
even
though it's
it’s exponential
exponential growth
growth..
though
Part 22 Solution:
data point
point farthest
farthest away
from the
best fit is at
at
Solution: From
From the
the scatterplot,
scatterplot, we
we can
can see
see that
that the
the data
away from
the line
line of best
I I
liters per
minute .
118 along
uptake of 2.s liters
along the
the x-axis.
x-axis. The
The point
point represents
represents an
an oxygen
oxygen uptake
per minute.
Note
point is likely an
hea vily influence
the line
our
Note that
that this
this data
data point
an outlier,
outlier, which
which can
can heavily
influence the
line of best
best fit and
and throw
throw off o
ur
predictions . Outliers
if they
they represent
represent specia
cases or exceptions.
exceptions.
predictions.
Outliers should
should be
be removed
removed from
from the
the data
data if
speciall cases
Not
the line
line of best
best fit, but
but you
' ll also
also be
be asked
interpret
N o t only
only will
will you
you be
be asked
asked to make
make predictions
predictions using
using the
you’ll
asked to
to interpret
its
We'll use
example in the
the next
next one
one to show
these concepts
its slope
slope and
and y-intercept.
y-intercept. We’ll
use the data
data from
from this
this example
show you
you how
how these
concepts
are tested.
are
tested .
255
CHAPTER
CHAPTER 28 STATISTICS
STATISTICS II
II
EXAMPLE
3:
Exam
3:
Oxygen Uptake versus Heart Rate
• _.Al
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110
120
100
110
Heart rate
rate(beats
per
minute)
H~
(beats per minute)
130
The saatterplot above shows the relationship between heart rate and oxygen uptake at
Themmwowdwwsflerdafimhipbetweenheanfitemdoxygenuptake
16dil‘ferentpoints
at 16
different points
dllring Kyle's exerciseroutine. The
vdminglfiyle‘sexemisemtine.
Thelineothestfitisalsoshown.
line ofhest fit is al.so shown.
PART'l:Which, of th following is the best-interpretation of the slope .of the line of best fit in the context
“M1:Winchofthefollowmgisthebestintapretahonotflieslopeoffllehneofbestfitinthecontext
ofthispmblem?
of this problell\'?
A) ‘I'hepredrctedmcreasemele’soxygenuptake
inlitersperminute,
foreveryonebeatperminute
The predicted increase in Kyle's oxygen uptake, in
liters per minute, for
every one beat per min~te
in his heart,:rate
' increase
incrmeinhishemme
B)
The predicted
B) The
predicted increase
increase'in
in Kyle's
Kyle's heart
heart rate,
rate, in beats
beats per
per minute,
for every
liter per
per minute
minute
minute, for
every one
one liter
jnctease
oxygen uptake
increasem
in his
his oxygen
uptake
‘ C) Kyle's
Kyle'spredicted
oxygen uptake
uptake in
per minute
minute at
heart rate
rate of
beats per
minute
predicted oxygen
in liters
lifers per
at a
a heart
of 0
Obeats
per minute
predicted heart
heart rate
rate in
in beats
beats per
per minute
minute at
at an
an oxygen
uptake of
liters per
D) Kyle’s
Kyle's predicted
oxygen uptake
of 0
Oliters
per minute
minute
PART
Which of the
following is the
the best
best interpretation
interpretation of
y-intercept of
of the
the line
line of
fit in
in the
the
PART 2: Which
the following
of the
they-intercept
of best
best fit
commit
of this
problem?
context-of
this problem?
A) The
predicted increase
in Kyle’s
oxygen uptake,
uptake, in
in liters
liters per minute,
for every
one beat
beat per minute
minute
A)
The predicted
increase in
Kyle's oxygen
minute, for
every one
increase
his heart
increase in his
rate
heart rate
B) the
13)
The _pre,djcted.increase
predictedi n c r e a s e in
in Kyle's
Kyle’s heart
heart rate,
rate, in
in beats
beats per
per minute,
liter per
per minute
minute
minute, for every
every one
pne liter
increase
:m,his
increasein
his oxygen
oxygen uptake
ptake
C)
Kyle's predicted
predicted oxygen
uptake in
in liters
liters per
per minute
minute at
heart rate
beats per
minute
C) Kyle's
oxygen uptake
at a
a heart
rate of 0
0 beats
per minute
Kyle’s predicted
predicted heart
rate in
in beats
beats per
per minute
minute at
at an
an oxygen
minute
D) Kyle's
heart rate
oxygen uptake
uptake of
of 0
0 liters
liters per
per minute
Part 1 Solution:
Solution: As we
we learned
model questions
the interpretation
chapter, the
slope is
is the
the
learned in the
the linear
linear model
questions in the
interpretation chapter,
the slope
increase
uptake)
for
each
increase
in
x
(heart
rate).
The
only
difference
n
o
w
is
that
it’s
a
predicted
increase in y31(oxygen
(oxygen uptake)
each increase
(heart rate) .
only difference now that it's predicted
[0]
increase.
increase . The
The answer
answer is (A) .
Part 2 Solution:
Solution: They
The y‐intercept
the value
value of y (oxygen
(oxygen uptake)
uptake) when
answer is
-intercept is the
when xx (the
(the heart
heart beat)
beat) is
is 0.
0. The
The answer
is
-.
Note that
value would
would have
have no
significance in
in real
real life since
heart rate
rate of
of 0.
[@.
Note
that this value
no significance
since you
you would
would be
be dead
dead at
at a
a heart
0.
This again
again illustrates
illustrates the danger
danger of predicting
values outside
outside the scope
sample data.
This
predicting values
scope of
of the
the sample
data .
256
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J A:,;i
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real estate age.ttt mtends to replicate th_r
sttrVey and ~ att~.mpt-to ~et ~ spiall~ marg:tn ofl~r ,,..._ ,
of the following samples will most .likely result in a sinallet margin of error for the mean pnce o.
apartment in Malden, Ma$sachuse.tts?
C) 80 randomly selected aparbnents in Malden
D) 80 randomly selected apartments in <lll.o{Massachusetts
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.The margin of error refers to the room for error we give to an estimate. For example, we
(3) S113MS119 3111
could say the mean price of an apartment in Malden is $150,000 with a margin of error of $10,000 . This implies
that the true mean price of all apartments in Malden is likely between $140,000 and $160,000 . This interval is
called a confidence interval (see Example 6).
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To get a smaller margin of error in Example 4, we should first only select from apartments in Malden. Selecting
apartments from all of Massachusetts not only introduces more variability to the data but also strays from the
original intent of the survey, which is to find the average price of Malden apartments . Secondly, we should use
a larger sample size. This is common sense. The more apartments we survey, the more accurate our data and
our estimations are and the lower our margin of error is.
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The larger the sample size and the less variable the data is, the lower the margin of error. We typically can't
control the standard deviation of the data (how spread out it is), but we can control the sample size. So why
don't researchers always use huge sample sizes? Because it's too costly and time-consuming to gather data
from everyone and everywhere .
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EXAMPtE5: :Researchers conducted/ an experimentio determine whether ex-e:rcise:im_pl'oves
, stuc(ent
exam scores. 'Fhey.z:a,:1domly
'S.elected.2()(1
.sfildents who exercise atrleastonce.'a week..ano
200 students
,t'he.
students' acacleinicpetform~-ees ~lor a .year
1
who do .not exercise at1east once-a week. Af:tertraciking
the researchers found tnat-the students ·who exercise at1east'Ohee a week pedo.rfned s1gnifitant1y better
on .fh:e,same exams thari the students who do not. l3ased.on the design.and.resrilts-..ofth-e
stud.y, which of
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the following is an appropriate conclusion?
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A) Exercising at least once a week is lik~y toimprove exam scores..
B) Exercising three times a week improves exam-scores more .than exen:isingjustonceaw.eek.
C) Any student who starts exercising at least once a week will improve his or her exam scores,
1
D) There is a positive association between exercise and student exam scores.
·1.·
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This question deals with a classic case of association (also called correlation) vs. causation . Just because
students who exercise got better exam scores doesn't mean that exercise causes an improvement in exam
scores. It's just associatedwith an improvement in exam scores. Perhaps students who exercise just have more
discipline or they have more demanding parents who make them study harder. Due to the way the experiment
was designed, we can't tell what the underlying factor is.
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Therefore, answer (A) is wrong because it implies causation. Answer (8) is wrong because it not only implies
causation but also implies that the frequency of exercise matters, something that wasn't tracked in the experiment.
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The answer is
)
257
CHAPTER 28
CHAPTER
28 STATISTICS
STATISTICS IT
[1
Answer
wrong because
outcome. Even
DID
improve exam
Answer (C)
(C) is wrong
because it suggests
suggests aa completely
completely certain
certain outcome.
Even if exercise D
I D improve
exam
scores,
improve their scores. There
There might
might be
students for
scores, not
n o t every
every single
single student
student who
who starts
starts exercising
exercising will
will improve
be students
whom
their scores worse.
worse . Any conclusion
drawn from sample
a generalization
and
whom exercising
exercising makes
makes their
conclusion drawn
sample data
data is a
generalization and
should
a truth
truth for every
should not
n o t be
be regarded
regarded as
asa
every individual
individual..
answer is ~-(D) . There
There is a positive
positive association
association between
between exercise and
and student
student exam
The answer
exam scores.
One of the
things the researchers
did correctly
correctly was
was to take
take random
random samples
each group.
word is
One
the things
researchers did
samples from each
group. The key word
random.. If
[f the samples
samples weren't
weren’t random,
random, we
we wouldn't
wouldn’t even
even have
have been
conclude that
there is
is aa positive
random
been able
able to conclude
that there
positive
association between
between exercise and
and exam
exam scores. Why? Let's
Let’s say the researchers
picked 30 students
from the
association
researchers picked
students from
the
tennis
team for the
the exercise group
group and
and 30 students
students who
who just
just play
video games
day for the non-exercise
non-exercise
tennis team
play video
games all day
group.. Definitely
Definitely not
n o t random
random.. Now,
Now, did
did the
the exercise group
group do
better on
their exams
because they
group
do better
on their
exams because
they exercise
or because
play tennis?
tennis? Or was
was it the
the video
video games
games that
that made
made the non-exercise
worse?
because they play
non-exercise group
group perform
perform worse?
selection wasn't
wasn’t random,
random, we
we can't
can’t tell how
how each
each factor influences
result.. When
the selection
Because the selection
influences the result
When the
selection is
random, all the
the factors
factors except
except the
the one we're
we’re testing
testing are
are "averaged
“averaged out.”
random,
out."
N o w what
what if the
the researchers
researchers wanted
wanted to see whether
whether exercise does
causee an
an improvement
does indeed
indeed caus
improvement in exam
exam scores.
Now
What should
should they have
have done
done differently?
differently? The answer
answer is random assignment.
Instead of randomly
selecting 200
assignment. Instead
randomly selecting
What
students from
from one
group that
that already
already exercises regularly
regularly and
and 200 students
another group
group that
that does
not,
students
one group
students from another
does not,
they should
step would
would be
be to randomly
assign each
each student
should have
have just randomly
randomly selected
selected 400 students
students.. The
The next
next step
randomly assign
student
to exercise or not.
least once a week
week and
everyone in the
not. Everyone
Everyone in the exercise group
group is forced to exercise at least
and everyone
the
group performs
performs better
better on the
exams, then
then we can
non-exercise
non‐exercise group
group is not
n o t allowed
allowed to exercise. If
If the
the exercise group
the exams,
can
conclude
conducting this
type of experiment
experiment
conclude that
that exercise causes
causes an
an improvement
improvement in exam
exam scores. Of course,
course, conducting
this type
can be
be such
such a monumental
monumental task.
task .
be extremely
extremely difficult, which
which is why proving
proving causation
causation can be
The following
following list summarizes
different experimental
experimental designs
designs involving
summarizes the
the conclusions
conclusions you
you can
can draw
draw from different
involving
two
exam scores).
two variables
variables (e.g.
(e.g. exercise and
and exam
1. Subjects not
not selected
selected at random
random & Subjects not
n o t randomly
randomly assigned
1.
assigned
• Results
0
Results cannot
cannot be
be generalized
generalized to the population
population..
Cause and
and effect cannot
cannot be
be proven.
proven.
•0 Cause
Example: Researchers
Researchers want
want to see whether
whether medication
medication X is effective in treating
•o Example:
treating the flu. People
People
with
the flu from Town A receive
People with
with the flu from Town B receive
a placebo
with the
receive medication
medication X. People
receive a
placebo
(sugar pill). More
More people
people in the
the medication
medication X group
group experience
symptoms. The
(sugar
experience a reduction
reduction in flu symptoms.
generalization
a reduction
reduction in flu symptoms
generalization that
that medication
medication X is associated
associated with
with a
symptoms cannot
cannot be
be made
made
since it was
only
tested
in
Town
A
and
Town
B
(sample
was
not
randomly
selected
from
the
general
was
tested Town and
(sample was not randomly selected
the general
population). There
There may
may be
be something
something special
special about
B. No cause
and effect
population).
about Town A and
and Town 8.
cause and
relationship
be established
established because
because the
the medication
medication was not
n o t randomly
assigned. Perhaps
Perhaps Town
relationship can be
randomly assigned.
experienced aa less severe
severe flu epidemic.
epidemic.
A experienced
not selected
selected at random
random &
& Subjects randomly
randomly assigned
assigned
2. Subjects not
Results cannot
cannot be
be generalized
generalized to
to the population.
population.
•0 Results
I
Cause and
and effect can
can be
proven.
•0 Cause
be proven.
Example: Researchers
Researchers want
want to
to see whether
whether medication
medication X
X is
is effective
in treating
flu. People
People
•0 Example:
effective in
treating the
the flu.
with the
the flu
flu from
from Town A
A and
and Town
To w n B
B are
are randomly
randomly assigned
medication X
placebo
with
assigned to
to either
either medication
X or
or aa placebo
(sugar pill).
pill). More
More people
people in
in the
the medication
medication X group
group experience
in flu symptoms.
symptoms. The
(sugar
experience aa reduction
reduction in
generalization that
that medication
medication X
X is
is effective
effective for
for everyone
be made
made since
since it
it was
was only
tested
generalization
everyone cannot
cannot be
only tested
in
Town A
A and
and Town
Town B
B (sample
(sample was
was n
o t randomly
randomly selected
selected from
general population).
population). Perhaps
in Town
not
from the
the general
Perhaps
only
one pa~cular
particular strain
strain of
of the
the flu_
flu exi~ts
exists in
in Town
Town A
A and
and Town
Town B.A
effect relationship
relationship
B. A cause
cause and
and effect
only one
can
be
established
because
the
medication
was
randomly
assigned.
For
the
people
in
Town
A and
and
can be established because the medication was randomly assigned. For the people in Town A
Town B,
B, we
we can
can c_
conclude
that medication
medication X
X causes
causes a
symptoms.. Note
that this
this is
T~w~
onc_lude that
a reduction
reduction in
in flu
flu symptoms
Note that
is
still
just
a
generalization‐as
with
any
other
medication,
medication
X
does
not
guarantee
you
will
s~ J~St a generalization-as with any other medication, medication X does not guarantee you will
definitely get
get better,
better, even
even if
if you
you live
live in
in Town
Town A
A or
or Town
B.
Town B.
definitely
258
I'
I
THE COLLEGE
COLLEGE PANDA
THE
PANDA
Subjects selected
selected at random
random &
& Subjects
Subjects not
not randomly
randomly assigned
assigned
3. Subjects
a
Results can
can be
be generalized
generalized to
to the
the population.
population.
• Results
Cause and
and effect cannot
cannot be
be proven.
proven.
•0 Cause
Example:: Researchers
Researchers want
w a n t to see
see whether
whether medication
People
•0 Example
medication X
X is effective
effective in treating
treating the
the flu. People
with
the
flu
from
the
general
population
are
randomly
selected.
They
are
given
the
choice
with the flu from the general population are randomly selected . They are given the choice of
of aa
n
e
w
medication
(medication
X)
or
a
traditional
medication
(really
a
sugar
pill).
More
people
in
the
new medication (medication X)
a traditional medication (really sugar pill). More people
the
medication
X
group
experience
a
reduction
in
flu
symptoms.
We
can
generalize
that
people
who
group experience a reduction
symptoms . We can generalize that people who
medication
choose to receive
receive medication
medication X fare
fare better
better than
However, no
cause and
than those
those who
who don’t.
don't. However,
no cause
and effect
choose
relationship can
can be
be established
established because
because the
the medication
o t randomly
don’t know
relationship
medication was
was nnot
randomly assigned.
assigned . We
We don't
know
whether the
the reduction
reduction in symptoms
symptoms is due
due to the
medication or a
between those
whether
the medication
a difference
difference between
those who
who
volunteered and
and those
those who
who didn't.
didn’t.
volunteered
4. Subjects
Subjects selected
selected at random
random &
Subjects randomly
randomly assigned
assigned
4.
& Subjects
Results can
can be
be generalized
generalized to the
the population.
population.
•0 Results
and effect can
can be
be proven
proven..
•0 Cause
Cause and
Example: Researchers
Researchers want
w a n t to see
see whether
whether medication
treating the
People
•0 Example:
medication X is effective
effective in treating
the flu. People
with the flu from
from the
the general
general population
population are
are randomly
randomly selected
selected.. Using
or tails),
with
Using aa coin
coin toss
toss (heads
(heads or
researchers randomly
randomly assign
each person
person to either
(sugar pill).
pill). More
More
researchers
assign each
either medication
medication X or a
a placebo
placebo (sugar
people
reduction in flu symptom
s. We
conclude that
people in the
the medication
medication X group
group experience
experience aa reduction
symptoms.
We can
can conclude
that
conclusion can
the entire
medication
medication X
X causes
causes aa reduction
reduction in flu symptoms
symptoms.. This conclusion
can be
be generalized
generalized to the
entire
population
with the
population of people
people with
the flu.
EXAMPLE
are testing pH levels
in a forest that is being hallned by acid rain.
They
EXAMPLE 6:
6:Environmentalists
Environmentalistsaretestingpfl
levelsinaforestthatisbeinghamedbyacid
rain. They
year and
and found
thatthe
analy.z.ed
from 40
rainfalls in the
analyzed water
water samples
samples from
40rainfallsin
the past
past year
found that
the mean.pH
mean pH of the-water
the water samples
samples
has
a 95% confidence
appropriate
has a
confidence interval
interval of 3.2 to 3.8. Which
Which of the
the following
following ~onclusions
conclusions is :the
the most
most appropriate
based
based on
on the
the confiaence
confidence interval?
interval?
A) 95% of all
the forest
have a
a pH
between 3.2 and
all the
forest rainfalls
rainfalls in
in the
the past
past year
year have
pH between
and 3.8.
B) 95% of all
a pH
~d 3.8.
all the
the forest
forest rainfalls
rainfalls in the
the past
past decade
decade have
have a
pH between
between 3.2 and
C)
!tis
plausible
that
the
true
mean
pH
of
all
the
forest
rainfalls
in
the
past
is between 3.2
and 3.8.
C) It is plausible that the true mean pH all the forest rainfalls the past year
yearisbetween
3.2 and
D)
is plausible that the true mean pH of
all the forest rainfalls in the past decade is between 3.2 and
D) It
ltisplausiblethatthetmemeaan
ofallthefomstrainfallsinthepastdecadeisbetweenB.2and
3.8.
If
don't know
't worry.
worry . You’ll
You'll never
and the
If you
you don’t
know what
what aa confidence
confidence interval
interval is, don
don't
never need
need to calculate
calculate one
one and
the SAT
makes
these
questions
very
easy.
All
a
confidence
interval
does
is
tell
you
where
the
true
mean
(or
some
makes these questions very
A l l a confidence interval does
you where the true mean (or some other
other
statistical
between 3.2 and
and 3.8). Even
Even though
the SAT only
statistical measure)
measure) for the
the population
population is likely
likely to be
be (e.g. between
though the
only
brings
99% (any
confidence intervals.
higher
brings up 95%
95°/o confidence
confidence intervals,
intervals, there
there are
are 97% and
and 99°/o
(any percentage)
percentage) confidence
intervals. The
The higher
the
the interval.
interval. 50
So in the
the example
example above,
above, we can
be
the confidence,
confidence, the more
more likely
likely the
the true
true mean
mean falls within
within the
can be
quite
rainfalls in the
between 3.2 and
quite confident
confident that
that the
the true
true mean
mean pH of all the
the forest
forest rainfalls
the past
past year
year is between
and 3.8. Answer
Answer
~(C) -. The
conclusions about
about the
the samples
The answer
answer is not
n o t (D) because
because we
we cannot
cannot draw
draw conclusions
the past
past decade
decade when
when all the
samples
w
e r e gathered
gathered from
from the
the past
year.
were
past year.
A confidence
rainfalls themselves.
themselves. You cannot
cannot say
one
confidence interval
interval does
does NOT
N O T say
say anything
anything about
about the
the rainfalls
say that
that any
any one
rainfall
has aa 95%
and 3.8, and
and you
you cannot
that 95% of all
rainfall has
95"/o chance
chance of having
having aa pH between
between 3.2 and
cannot say
say that
all the
the forest
forest
rainfalls
remember that
confidence interval
interval applies
applies
rainfalls in the
the past
past year
year had
had a pH between
between 3.2 and
and 3.8. Always
Always remember
that a confidence
only
the mean,
an individual
individual data
group of data
only to the
mean, which
which is a statistical
statistical measurement,
measurement, NOT
NOT an
data point
point or a group
data
points.
points.
Secondly,
there is a
a 95% chance
chance it contains
true mean.
mean. Even
Secondly, aa 95% confidence
confidence interval
interval does
does not
n o t imply
imply that
that there
contains the
the true
Even
though confidence
you cannot
cannot say
say that
that the
has a
a 95%
though
confidence intervals
intervals are
are computed
computed for the
the mean,
mean, you
the interval
interval of 3.2 to 3.8 has
chance
containing the
m e a n pH
pH..
chance of containing
the true
true mean
259
II
STATISTICS II
CHAYTER 28 STATISTICS
CHAPTER
and
again and
repeated again
were repeated
experiment were
If the experiment
confident in something?
be 95% confident
statistics to be
mean in statistics
does itit mean
So what does
Sowhat
something? If
contains
that
interval
confidence
a
us
give
would
experiments
those
of
95%
samples,
water
40
with
each
again,
again, each with 40 water samples,
those experiments would
us a confidence interval that contains
experiment.
one experiment.
result of just one
the result
example is the
given in the example
interval given
confidence interval
words, the confidence
other words,
mean. In other
true mean.
the true
interval. Keep
different confidence
a different
produce a
would produce
samples) would
40 samples)
(another 40
experiment (another
same experiment
Another rrun
Another
u n of the same
confidence interval.
Keep
pertains to all
the 95% pertains
So the
mean. 50
true mean.
contain the true
will contain
them will
and 95% of them
intervals and
confidence intervals
these confidence
getting these
on getting
interval
confidence interval
one confidence
any one
that any
chance that
NOT the chance
experiments, NOT
repeated experiments,
generated by repeated
intervals generated
confidence intervals
the confidence
that
aware that
be aware
but be
calculated, but
are calculated,
intervals are
confidence intervals
how confidence
about how
worry about
don't worry
mean . Again,
true mean.
contains the true
contains
Again, don’t
statistics.
how "confidence"
this is how
”confidence” is defined
defined in statistics.
260
260
THE
COLLEGE PANDA
THE COLLEGE
PANDA
CHAPTER EXERCISE:Answers for this chapter start on page 340.
A calculator
calculator is allowed
allowed on the following
following
questions.
questions.
"'
§
·.c
ro
Violations in Various
Towns
Traffic Light
Light Violations
Various Towns
0
·;;: 100
.....
Tu 90
Male Shoe Size versus Age
13
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12
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70
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10
10 11
11 12
12 13
13 14
14 15
15 16
16 17
17 18
18 19
19 20
20
Age
Age (years)
(years)
20
20
30
30
<lJ
~
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50
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/
.........
60
•.----
-
:
-
-
40
600 70
800 9
900 1 100
4 0 50
50 6
70 8
00
Number of traffic lights
Number
lights
The scatterplot
above shows
shows the
the number
The
scatterplot above
number of
lights in 15
and the
the average
weekly
traffic lights
15 towns
towns and
average weekly
number
number of traffic light
light violations
violations that
that occur
occur in
each town.
town . The
line of best
fit is also
also shown.
each
The line
best fit
shown.
the line
best fit, which
which of the
the
Based on the
line of best
following
weekly
following is the
the predicted
predicted average
average weekly
number of traffic light
light violations
a town
number
violations in a
town with
with
75 traffic lights?
75
lights?
The scatterplot
shows the
the relationship
scatterplot above
above shows
relationship
between
between age,
age, in years,
years, and
and shoe
shoe size
size for 24
24 male
maless
between
and 20
between 10
10and
20 years
years old.
old. The
The line
line of best
best fit
data, how
is also
also shown.
shown. Based
Based on the
the data,
how many
many 19
19
year
year old
old males
males had
had aa shoe
shoe size
size greater
greater than
than the
the
one predicted
predicted by
one
by the
the line
line of best
best fit?
A ) 1
A)
B
B)) 2
A
0
A)) 440
C)3
C)
B) 50
50
D ) 44
D)
C
5
C)) 555
D
D)) 6600
In a survey
s, xatpercent
percent said
survey of 400 senior
seniors,
said that
that
they
physics. One
they plan
plan on
on majoring
majoring in physics.
One university
university
has
this data
has used
used this
data to estimate
estimate the
the number
number of
physics
major s it expects
physics majors
expects for its
its entering
entering class
class of
3,300 students
the university
students.. If
If the
university expects
expects 66
66
physics
physics majors,
majors, what
what is the
the value
value of x ??
261
CHAPTER 28 STATISTICS II
A university
wants to determine
university wants
determinethe dietary
dietary
preferences of the
freshman class.
preferences
the students
students in its
its freshman
Which of the
the following
survey methods
methods is
is most
most
Which
following survey
likely
provide the
most valid
results?
likely to provide
the most
valid results?
Consumer Behavior
Behavior during
Consumer
during Store
Store Sales
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l
!
50
- 45
40
..... 35
~ 30
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0.. 25
0..
0
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15
~ 10
A) Selecting
Selecting aa random
random sample
sample of 600 students
students
from
the university
from the
university
I
.§
B) Selecting
Selecting aa random
random sample
sample of 300 students
students
from
from the
the university's
university’s freshman
freshman class
class
C) Selecting
Selecting aa random
random sample
sample of 600 students
students
the university’s
university's freshman
freshman class
from the
class
D) Selecting
Selecting aa random
random sample
sample of 600 students
students
from one
one of the
the university's
university’s freshman
freshman
dining
dining halls
halls
=
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5
0
0
0 55 110
0 115
5 220
0 225
5 330
0 335
5 440
0 445
5 550
0
(%}
Store Discount
Store
Discount (%)
Shopping time
time refers
the time
time a
a customer
Shopping
refers to the
customer
spends
in
one
store.
The
scatterplot
shows
spends one store. The scatterplot above
above shows
the average
average shopping
shopping time,
time, in minutes,
the
minutes, of
customers at 26 different stores
stores offering
offering various
various
customers
discounts. The
The line
line of best
also shown
discounts.
best fit is also
shown..
Which of the following
following is the
interpretation
Which
the best
best interpretation
meaning of the
the y-intercept
the line
line of
of the meaning
y-intercept of the
best fit?
best
Two candidates
candidates are
are running
running for governor
governor of aa
state.
that out
a random
state. A recent
recent poll
poll reports
reports that
out of a
random
sample
sample of 250 voters,
voters, 110 support
support Candidate
Candidate A
and 140 support
Candidate B. An estimated
and
support Candidate
estimated
500,000 state
expected to vote
state residents
residents are
are expected
vote on
election
election day. According
According to the
the poll,
poll, Candidate
Candidate B
is expected
expected to receive
receive how
how many
more votes
many more
votes
than Candidate
Candidate A?
than
A) The predicted
predicted average
average shopping
time, in
shopping time,
minutes, of customers
store offering
offering no
minutes,
customers at a store
discount
discount
A) 60,000
shopping time,
B) The predicted
predicted average
average shopping
time, in
minutes, of customers
customers at a store
offering a
minutes,
store offering
discount
50% discount
B) 130,000
C) 220,000
D) 280,000
C) The
predicted increase
the average
C)
The predicted
increase in the
average
shopping
time, in minutes,
each one
shopping time,
minutes, for each
one
percent
the store
store discount
discount
percent increase
increase in the
D) The predicted
number of
predicted average
average number
customers at a store
offering no discount
customers
store offering
discount
262
THE COLLEGE
COLLEGE PANDA
PANDA
Advertising for 16 Companies
,....._
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~ 450
~ 400
~ 350
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~ 300
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ro
Movie Length versus Box Office Sales
•
•
•
- .
0
0
1 0 20
2 0 30
3 0 4400 550
0 60
6 0 70
7 0 80
8 0 90
90100
0 10
100
Advertising Expenses
Expenses (in
(in thousands
dollars)
Advertising
thousands of dollars)
The scatterplot
scatterplot above
above shows
shows the
the relationship
relationship
between revenue
between
revenue and
and advertising
advertising expenses
expenses for
16
companies . The line
16companies.
shown.
line of best
best fit
fit is also
also shown.
Which of the
the following
following is the
the best
best interpretation
Which
interpretation
meaning of the
slope of the
the line
of the
the meaning
the slope
line of best
best fit?
><
0
c:o
.,_
_ ,
•
. . . . . . . . ~ _
o ~~--+--~---~~-~-~~
60 70 80 90 100 110 120 130 140 150
Movie
Movie Length
Length (minutes)
(minutes)
The scatterplot
15
scatterplot above
above plots
plots the
the lengths
lengths of 15
movies against
movies
sales. The
The line
line of
against their
their box
box office sales.
best fit is
best
is also
also shown.
shown. Which
the following
following is
is
Which of the
the best
the
best interpretation
interpretation of the
the meaning
meaning of the
the
slope
slope of the line
line of best
best fit?
expected increase
increase in revenue
A) The expected
revenue for every
every
one dollar
dollar increase
one
increase in advertising
advertising expenses
expenses
B) The expected
expected increase
increase in revenue
revenue for every
every
one
one thousand
increase in advertising
advertising
thousand dollar
dollar increase
A) The expected
expected decrease
box office sales
decrease in box
sales
per
increase in movie
movie length
per minute
minute increase
length
expected increase
increase in box
B) The
The expected
box office sales
sales per
per
minute increase
minute
increase in movie
movie length
length
C) The expected
C
sales
expected decrease
decrease in box
box office sales
per
movie length
length
per 10‐minute
10-minute increase
increase in movie
D) The
The expected
expected increase
increase in box
box office sales
sales per
per
10-minute
10‐minute increase
increase in movie
movie length
length
expenses
expenses
C)
C) The
The expected
expected increase
increase in advertising
advertising
expenses for every
expenses
thousand
dollar
every one
one thousand dollar
increase
increase in revenue
revenue
D) The
The expected
expected revenue
company that
that
revenue of a
a company
D
has
no advertising
has no
advertising expenses
expenses
263
CHAPTER 28 STATISTICS
CHAPTER
STATTSTTCS II
II
Mistakes Made
Made in Incentive-based
Mistakes
Incentive-based Task
60
60
(I)
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(ll
E
50
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(I)
40
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100
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.0
0
Fat and Calories of Ice Cream
,,
l
200 300 400
Prize ((in
Prize
i n dollars)
dollars)
500
300
600
a psychological
psychological study,
In a
study, researchers
researchers asked
asked
participants
each complete
participants to each
complete aa difficult
difficult task
task for
/
V
I
15
20
20
25
30
30
(grams)
Total fat (grams)
35
The
The scatterplot
above shows
shows the
content and
scatterplot above
the fat content
and
calorie counts
counts of 8 different
calorie
different cups
cups of ice cream.
cream .
Based on the line
Based
line of best
best fit to the
the data
data shown,
shown,
what is the
the expected
what
expected increase
in the
number of
increase in
the number
of
calories for each
calories
of
fat
in
a
cup
each additional
additional gram
gram
in a cup
of ice cream?
cream?
a cash
cash prize,
prize, the
the amount
amount of which
a
varied from
from
which varied
participant to participant.
participant
participant. The
The results
results of the
the
study, as
as well
well as
as the
the line
line of best
study,
best fit, are
are shown
in
shown in
the scatterplot
above. Which
the
scatterplot above.
Which of the
following is
the following
the best
best interpretation
interpretation of the
the
the meaning
meaning of the
the
y-intercept of the
y-intercept
line of best
the line
best fit?
A) 5
B) 8
The expected
expected decrease
A) The
number of
of
decrease in the
the number
C) 20
20
D
)
4
0
D) 40
mistakes made
mistakes
made per
the
per dollar
dollar increase
increase in the
cash
cash prize
prize
B) The
The expected
expected increase
increase in the
the number
number of
of
mistakes made
mistakes
made per
per dollar
dollar increase
increase in the
the
cash
cash prize
prize
C) The
C
expected dollar
cash
The expected
dollar amount
amount of
of the
the cash
prize required
prize
required for a
a person
person to complete
complete the
the
task
task with
with 0 mistakes
mistakes
The expected
expected number
D) The
number of mistakes
mistakes aa person
person
makes in completing
makes
completing the
the task
task when
when no
no cash
cash
prize
prize is offered
offered
A record
record of driving
driving Violations
violations by type
type and
and
vehicle is shown
below.
vehicle
shown below.
‐
m
”
Truck
Car
Total
Speeding
speeding
68
83
151
Violation
Violation Type
Stop
Sign Parking
stop Sign
17
39
51
26
90
43
m
m
-
Total
124
124
160
If
violation
If the data
data is used
used to estimate
estimate driving
driving violation
information
information about
about 2,000 total
total violations
violations in
in a
a
certain
the following
best
certain state,
state, which
which of
of the
following is
is the
the best
estimate of the number
estimate
number of speeding
speeding violations
violations
committed
committed by cars in the state?
state?
A)
A) 479
B) 585
C) 1063
D) 1099
264
u
284
THE
THE COLLEGE
COLLEGE PANDA
PANDA
Nitrogen Fertilizer and Oats
,,...._80
~
~
•
~ 60
en
~ 50
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en
40
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ro
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60
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en
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QJ
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>=
Food Courts in Various Malls
100
en
I
70
....
~::i
0
0
,
0
200
200
300
400
100
300
400
500
Amount
Amount of nitrogen
nitrogen applied
applied (pounds
(pounds per
per acre)
acre)
~
ro
...
QJ
~
,i
!
r---
20 -0
4
5
6
7
8
9
10
Number
restaurants
Number of restaurants
The
scatterplot above
above shows
shows the
distribution of
The scatterplot
the distribution
seats for the
different mall
food
seats
the restaurants
restaurants in 7 different
mall food
courts. The
line of best
best fit is also
also shown.
courts.
The line
shown.
According
data , what
what is
total number
According to the
the data,
is the
the total
number
at the
the food
court represented
the data
data
of seats
seats at
food court
represented by
by the
point
that is farthest
farthest from
the line
point that
from the
line of best
best fit?
The
The scatterplot
scatterplot above
above shows
shows the
the amount
amount of
nitrogen
oat fields
nitrogen fertilizer
fertilizer applied
applied to
to 8 oat
fields and
and
their
best fit is
their yields
yields.. The
The line
line of best
is also
also shown
shown..
Which
the following
is closest
Which of the
following is
closest to
to the
the amount
amount
of nitrogen
per acre,
nitrogen applied,
applied, in pounds
pounds per
acre, to the
the
oat
oat field
field whose
whose yield
yield is
is best
best predicted
predicted by
by the
the line
line
of best
best fit?
A) 200
B)
B)
C)
C)
A)
A) 200
B) 350
C)
400
C) 400
240
240
320
320
D) 560
D)
D) 450
Researchers
an experiment
Researchers must
must conduct
conduct an
experiment to see
see
whether a
an
new
vaccine is effective
effective in relieving
relieving
whether
e w vaccine
certain
have selected
random
certain allergies.
allergies . They
They have
selected a
a random
sample
100 allergy
Some of the
sample of 100
allergy patients.
patients. Some
the
patients are
are assigned
assigned to
the new
vaccine while
while
patients
to the
n e w vaccine
the rest
rest are
are assigned
assigned to
to the
the traditional
treatment.
the
traditional treatment.
Which of the
following methods
Which
the following
methods of assigning
assigning
each patient’s
patient's treatment
likely to lead
lead to
each
treatment is
is most
most likely
to
a reliable
reliable conclusion
conclusion about
about the
the effectiveness
a
effectiveness of
the
e w vaccine?
the n
new
vaccine?
assigned to the
new
A) Females
Females are
are assigned
the n
e w vaccine.
vaccine.
Those who
more than
than one
one allergy
allergy are
B) Those
who have
have more
are
assigned
n e w vaccine.
assigned to the
the new
vaccine .
C) The
divide themselves
evenly into
into
The patients
patients divide
themselves evenly
ttwo
w o groups.
groups. A coin
coin is tossed
tossed to decide
decide
which
group receives
receives the
e w vaccine.
which group
the nnew
vaccine.
D
patient is assigned
assigned aa random
random number.
D) Each
Each patient
number.
Those with
even number
are assigned
assigned to
Those
with an
an even
number are
the
e w vaccine.
vaccine.
the n
new
265
CHAPTER 28
CHAPTER
28 STATISTICS
II
STATISTICS II
The
blue-spotted salamander's
salamander’s tail
tail
The length
length of a
a blue-spotted
can
its age.
A biologist
biologist
can be
be used
used to estimate
estimate its
age. A
selects
random
selects 80blue-spotted
80 blue-spotted salamanders
salamanders at
at random
and
and finds
finds that
that the
the average
average length
length of their
their tails
tails
has
has a
a 95°/o
95% confidence
confidence interval
interval of 5 to 6 inches.
inches.
Which
Which of the
the following
following conclusions
conclusions is the
the most
most
appropriate based
appropriate
based on
on the
the confidence
confidence interval?
interval?
basketball manufacturer
selects a
A basketball
manufacturer selects
a random
random
sample of its
sample
ensure aa
its basketballs
basketballs each
each week
week to ensure
consistent air
consistent
within them
air pressure
pressure within
them is
maintained. ln
maintained.
In Week 1,
1, the
sample had
had aa mean
mean
the sample
air pressure
pressure of 8.2 psi
air
(pounds per
per square
square inch)
psi (pounds
inch)
and a
a margin
margin of error
0.1 psi.
and
error of 0.1
psi. ln
In Week 2,
the
2, the
sample
sample had
mean air pressure
pressure of 7.7 psi
had aa mean
psi and
and a
a
margin of error
margin
error of 0.3 psi.
psi . Based
Based on
on these
these results,
results,
which of the
the following
reasonable
which
following is aa reasonable
conclusion?
conclusion?
A) 95°/o
salamanders have
have aa
95% of all
all blue-spotted
blue-spotted salamanders
tail that
between 5 and
and 6 inches
inches in length
length..
that is between
all salamanders
have aa tail
B) 95% of all
salamanders have
that is
tail that
between 5 and
between
length.
and 6 inches
inches in length.
The true
C) The
true average
the tails
average length
length of the
tails of all
all
blue-spotted salamanders
salamanders is
likely between
blue-spotted
is likely
between
Most of the
the basketballs
basketballs produced
produced in Week 11
A) Most
had an
an air
air pressure
pressure under
under 8.2 psi,
had
psi, whereas
whereas
most of the
most
the basketballs
basketballs produced
produced in Week 2
had
an air
had an
air pressure
pressure under
under 7.7 psi.
psi.
B) The
pressure of all
the basketballs
basketballs
The mean
mean air
air pressure
all the
produced in Week 11 was
was 0.5 psi
produced
psi more
more than
than
·tthe
mean air
pressure of all
the basketballs
h e mean
air pressure
all the
basketballs
produced
produced in Week 2.
2.
number of basketballs
basketballs in the
the Week 1
C) The
The number
sample
was more
more than
the number
number of
sample was
than the
basketballs in the
the Week 2 sample.
sample.
basketballs
that the
the mean
mean air
D) It is very
very likely
likely that
air pressure
pressure
the basketballs
produced in Week 11
of all
all the
basketballs produced
was less
less than
was
than the
all
the mean
mean air
air pressure
pressure of all
the
basketballs produced
the basketballs
2.
produced in Week 2.
and 6 inches.
5 and
inches.
The true
true average
average length
length of the
the tails
tails of all
D) The
all
salamanders
salamanders is likely
likely between
between 5 and
and 6
inches .
inches.
An economist
economist conducted
conducted research
determine
research to determine
whether there
there is a
a relationship
relationship between
between the
whether
the price
price
food and
and population
population density.
of food
density. He
He collected
collected
data from
from a
a random
random sample
data
S . cities
sample of 100 U
U.S.
cities and
and
found significant
significant evidence
found
food
evidence that
that the
the price
price of food
lower in places
places with
is lower
population
with a
a high
high population
density. Which
Which of the
density.
the following
conclusions is
following conclusions
best
by these
these results?
results?
best supported
supported by
A student
student is assigned
assigned to conduct
conduct a
a survey
survey to
determine the
the mean
number of servings
determine
mean number
servings of
vegetables eaten
eaten by aa certain
vegetables
group of people
certain group
people
each day
has not
yet decided
each
day.. The
The student
student has
n o t yet
decided which
which
group
group of people
people will
will be
be the
focus of this
this survey.
survey.
the focus
Selecting
Selecting aa random
random sample
sample from
from which
the
which of the
following
groups would
would m
o s t likely
likely give
following groups
most
give the
the
smallest
error?
smallest margin
margin of error?
ln US.
U.S. cities,
cities, there
A) in
there is a
positive association
association
a positive
between the
between
the price
price of food
food and
and population
population
density.
density .
U.S. cities,
cities, there
B) In US.
there is aa negative
negative association
association
between the
the price
between
price of food
food and
and population
population
density.
density.
C) In U
S . cities,
U.S.
cities, aa decrease
decrease in the
the price
price of food
food
is caused
an increase
increase in
in the
caused by
by an
the population
population
density.
density.
U.S.
D) In U
S . cities,
cities, an
an increase
increase in the
the population
population
density
price
density is caused
caused by a decrease
decrease in the
the price
of food.
food.
the same
A) Residents
Residents of the
same city
Customers of a certain
certain restaurant
B) Customers
restaurant
Viewers of the
the same
same television
television show
C) Viewers
show
Students who
are following
D) Students
who are
the same
same daily
daily
following the
diet
diet plan
plan
266
266
29
Volume
Volume
The volume
volume of all
all regular
regular solids
solids can
can be
be found
found using
using the
the following
following formula:
formula:
The
Volume = Area of base x height
3 (the
2 and
That’s why
why the
the volume
volume of aa cube
cube is V =
: s53
(the area
area of the
base is s52
and the
the height
height is s)
s)
the base
That's
sS
The volume
volume of aa rectangular
rectangular box/prism
box / prism is
is V
V=
: lwh
Izuh (the
(the area
area of the
base is lw
Iw and
and the
height is h)
h)
The
the base
the height
h
w
I
2 h (the
And
the base
and the
the height
h)
A n d the
the volume
volume of aa cylinder
cylinder is V
V=
: m
rrrzlz
(the area
area of the
base is
is m
7rr22 and
height is
is 11)
h
Even though
though the
the SAT gives
gives you
you these
these formulas
formulas at
at the
the beginning
beginning of each
each math
math section,
section, they
they should
should be
be memorized,
memorized,
Even
addition to
to the
the volume
volume of aa cone
cone
in addition
V : ‐3 rrrzlI
267
CHAPTER
VOLUME
CHAPTER 29 VOLUME
and the
and
the volume
volume of a
a sphere
sphere
4 3
V=
: -§nr3
V
rrr
3
But what
what if we
we have
What’s the
volume of
have a
a hollowed-out
hollowed-out cylinder?
cylinder? What's
the volume
of that?
that?
h
I
'
we look
Well, ifif we
look at
base, it's
it’s just
just a
ring.
at the
the base,
a ring.
A
area of the
The area
the outer
minus the
inner circle.
the ring
ring is the
outer circle minus
the inner
7tR2 ‐ 7tr2 = 7r(R2 ‐
r2)
To get
by the
the height.
get the
the volume,
vo lume, we
we multiply
multiply this
this area
area by
height.
v =
: rr(R
7r(R22 -‐ r2)h
V
r 2 )h
addition to finding
finding an
an object’s
object's volume,
In addition
volume, you'll
you'll also
also need
need to know
know how
how to find its density.
Sometimes you'
you’ll
density. Sometimes
ll
the density
be given
given the
density formula
and sometimes
sometimes you
you won't,
won’t, so
so it’s
it.
formula and
it's important
important to memorize
memorize
DeI\S;l
e ·ty i =
ns ty
=Mass‐ ‐
Volume
Volume
Denser objects are heavier
Denser
size.
heavier relative
relative to their
their size.
268
THE
COLLEGE PANDA
THE COLLEGE
PANDA
CHAPTER EXERCISE: Answers for this chapter start on page 342.
calculator is allowed
allowed on the following
following
A calculator
questions.
questions.
What is the
the vo
volume
cube with
area
What
lume of a
a cube
with surface
surface area
24a 2 ?
24a2
A) 4a2
4a2
A)
8a2
B) 8a2
,,
,
,
,
,,
,
C) 8a3
8a3
C)
I
I
3
D)
D) 16a
16113
5cm
I
water tank
tank with
with a
radius of 4
A cylindrical
cylindrical water
a base
base radius
4
feet
and a
a height
can be
filled in 33 hours
feet and
height of 6 feet can
be filled
hours..
At that
that rate,
rate, how
will it take
take to fill a
how many
many hours
hours will
cylindrical water
a base
cylindrical
water tank
tank with
with a
base radius
radius of 6
feet and
and a
a height
height of 8
8 feet?
feet?
4cm
In the
the figure
figure above,
above, aa cylindrical
cylindrical block
block of
of wood
wood
is sliced
sliced into
into two
two pieces
pieces as
as shown
shown by
by the
the dashed
dashed
curve.
curve. What
What is the
the volume
volume of the
the top
top piece
piece in
A) 4.5
cubic
cubic centimeters?
centimeters?
B) 6
A) lOrr
107i
C) 7.5
B) 15rr
157r
D) 9
2071
C) 207T
D) 407r
40n
A clay brick
shape of aa right
brick in the
the shape
right rectangular
rectangular
prism has
length of 6 inches,
inches, a width
prism
has aa length
width that
that is
25% greater
than
its
length,
and
a
height
greater than its length, and a height that
that is 2
inches shorter
shorter than
its length.
length. The
The brick
brick has
has aa
inches
than its
mass of 5.85 kilograms.
kilograms. What
What is
is the
the density,
density, in
mass
grams per
per cubic
inch, of the
the brick?
brick?
grams
cubic inch,
James
James wants
wants to
to cover
cover aa rectangular
rectangular box
box with
with
wrapping
has aa square
wrapping paper
paper.. The
The box
box has
square base
base with
with
an
an area
area of 25 square
square inches.
inches. The
The volume
volume of the
the
box
box is 100 cubic
cubic inches.
inches. How
H o w many
many square
square
inches
paper will
inches of wrapping
wrapping paper
will James
James need
need to
to
exactly
cover
all
faces
of
the
box,
including
exactly cover all faces the box, including the
the
top
top and
and the
the bottom?
bottom?
A) 120
B) 130
C) 150
D) 160
269
CHAPTER
CHAPTER 29 VOLUME
VOLUME
A cube
side length
length of 5 inches
cube with
with a
a side
inches is painted
painted
black
black on all
all six faces.
faces. The
The entire
entire cube
cube is then
then cut
cut
into smaller
smaller cubes
cubes with
with sides
sides of 1 inch.
inch. How
into
How
many
any black
black paint
many small
small cubes
cubes do
do n_ot
not have
have any
paint
them?
on them?
A)
A ) 227
7
B)) 3
311
B
C)) 336
C
6
D)) 4488
D
finds a
a box
box with
an open
open top.
top. Each
Each side
Yuna finds
with an
side is 8
inches long.
long . If she
this box
with identical
identical 2
inches
she fills this
box with
cubes, how
many of these
these
in by 2 in by 2 in cubes,
how many
cubes will
will be
be touching
touching the
the box?
box?
cubes
A container
container in the
the shape
shape of a right
right circular
circular
cylinder
shown
above
is
just
large
enough
cylinder shown above is just large enough to
to fit
exactly 33 tennis
exactly
tennis balls
balls each
each with
with aa radius
radius of 22
inches
emptied o
out
inches.. If
If the
the container
container were
were emptied
u t and
and
filled
top with
be the
filled to
to the
the top
with water,
water, what
what would
would be
the
volume
volume of water,
water, in cubic
cubic inches
inches,, held
held by the
the
container?
container?
A)) 440
A
0
B) 48
48
B)
C)) 552
C
2
D)) 556
D
6
A)
A) 16n
1671
B) 24n
247r
IN
C) 32n
327:
D)
D) 48n
487r
A 3 x 4 x><5Ssolidblockismadeupofl
solid block is made up of 1 xx11 x1
x1
A3><4
unit cubes.
cubes. The
unit
The outside
outside surface
surface of the
the block
block is
painted
black. How
How many
cubes have
have
painted black.
many unit
u n i t cubes
exactly
exactly one
one face painted
painted black?
black?
A ) 116
6
A)
An aquarium
aquarium has
inch
has an
an 80
80 inch
inch by 25
25 inch
rectangular
rectangular base
base and
and aa height
height of 30
30 inches
inches.. The
The
aquarium is filled with
aquarium
with water
water to
to aa depth
depth of 20
20
inches
block with
inches.. If
If aa solid
solid block
with a volume
volume of 5,000 in
in33
aquarium, by
is completely
completely submerged
submerged in the
the aquarium,
by
how
does the
water level
how many
many inches
inches does
the water
level rise?
rise?
B
8
B)) 118
C
0
C)) 220
D
2
D)) 222
right circular
circular cone
cone has
has aa volume
6rra 4 cubic
cubic
A right
volume of 67ra4
centimeters, where
positive constant.
constant. If
the
centimeters,
where a is a positive
If the
2
which of
height of the
the cone
cone is 2a
height
2112 centimeters,
centimeters, which
the following
following gives
give s the
the radius,
radius, in centimeters,
centimeter s, of
the
the base
base of the
the cone
?
the
cone in terms
terms of a ?
A)
A) tat/3
a./3
B)
B) 311
3a
C)
C) 3a2
3a 2
D)
D)
270
941
9a
THE
COLLEGE PANDA
THE COLLEGE
PANDA
3
10
15 cm
8
crate that
that is 10
inches long,
long, 8 inches
wide, and
and
A crate
10inches
inches wide,
3 inches
inches high
high is shown
above. The
and the
the
shown above.
The floor
floor and
four walls
walls are
are all one
one inch
inch thick.
thick. H
How
many
four
o w many
one-inch cubical
cubical blocks
can fit inside
inside the
the crate?
crate?
one-inch
blocks can
A food
produces packages
food manufacturer
manufacturer produces
packages of
frozen
cones. Each
frozen ice
ice cream
cream cones.
Each ice
ice cream
cream cone
cone
consists
that is filled
consists of aa right
right circular
circular cone
cone that
filled with
with
ice
hemisphere is formed
the
ice cream
cream until
until aahemisphere
formed above
above the
cone
cone as
as shown
shown in the
the figure
figure above.
above. The
The right
right
circular
has aa base
radius of 9 cm
circular cone
cone has
base radius
cm and
and aa
slant
height
of
15
cm
.
What
is
the
volume
slant height 15cm. What is the volume of ice
ice
cream,
cream, in cubic
cubic centimeters,
centimeters, the
the manufacturer
manufacturer
uses
uses for each
each ice cream
cream cone?
cone?
A)) 884
A
4
B) 96
B)
C) 120
D) 144
A)
A) 729n
7297r
B) 810n
8107r
C) 8917:
891 rr
96071
D) 960n
A right
right circular
circular cylinder
cylinder has
has aa base
base radius
radius rr that
that
is 22 inches
inches longer
longer than
than its
its height.
height. Which
Which of the
the
following expressions
expressions gives
gives the
the volume,
volume, in cubic
cubic
following
inches, of the
inches,
the cylinder
cylinder in terms
terms of rr ??
5
A)
271'r33
A) 2nr
6
Note: Figure
o t drawn
to scale
scale..
Note:
Figure n
not
drawn to
7rr3
+ 2n
2m2
B) n
r3 +
r2
The concrete
staircase shown
shown above
The
concrete staircase
above is built
built from
from
a
long and
a rectangular
rectangular base
base that
that is 55 meters
meters long
and 6
6
meters
have equal
equal
meters wide.
wide . The
The three
three steps
steps have
dimensions
one has
rise of 0.2 meters.
meters.
dimensions and
and each
each one
has a
a rise
If
is 130 kilograms
kilograms per
per
lf the
the density
density of concrete
concrete is
cubic
mass of the
concrete
cubic meter,
meter, what
what is the
the mass
the concrete
staircase
mass divided
staircase in kilograms?
kilograms? (Density
(Density is mass
divided
by volume)
volume)
by
r 3 -‐ 2n
r2
C) nm3
27rr2
D) 2n
27rr3
+ nm2
D)
r3 +
r2
A) 1,420
B) 1,560
C) 1,820
D) 2,040
271
271
30
the
Answers
to.the
Answers to
Exercmes
Exercises
l: Exponents & Radicals
Chapter
Chapter 1:Exponents
Radicals
EXERCISE
EXERCISE1:
1. 1
36
11. -~36
9
20. 9
2. -711
12. 64
1l
21. 9
21.
9
3. 1
72
13. -7‐72
4. -A11
14. 108
5. 11
648
15. -7648
1
23. ‑
23
· 125
125
6. -211
16. 11
24. 49
24- 49
7.
7. -‐1 1
1
17
17.. !6g
25_
*
25. 49
11
18. 4
7
18.
4
26. 1,000
26
- 27
8. 727
9. -727
27
10. 27
27
10.
22. 125
1
49
1
1
27
27
· 1,000
1,000
19. 1
272
THE
THE COLLEGE
PANDA
COLLEGE PAN
DA
EXE
EXERCISE
RCISE 2:
2:
1 6x
6x55
1.
88
2. k2
‐
2.
k2
3. 15x
15x22
4. -‐ 2211
4
10 x33
10.
x
19 36m8
19. 36m8
1
~1
20. ”6
20.
6
x6
X
11. y‐3
11.
3
y
6
a
21.
3112
1,12
b12
3u 2
12. T
4
22 m4
13. -‐8u3v3
13.
8113 v 3
22.' -n
n
21.
1114
11
5. 7
5.
8x 6
8x
9b55
6. -‐&a3
113
14 15
23.1’2
14. x 5
23. x 2
15. 3x
3x88
15.
24‘ L 1
24. -mn2
-2
16. Xx
25. k
25. k
mn
7
7.
n4
1
14
. 22
8.
8 crib6
a4b6
y22
17. xx99
17.
6
m6
26. ‑m9
"9
22
18. 3‐3
X
x
n
27 5 7 9
27.. x5yy7zZ9
9. 2
y_2
9.
X
x
EXE
EXERCISE
3:
RCISE 3:
1·
Ni
l.1. 2/3
7. 4\/'2
Ni
7.
2. 4/6
4\/6
2.
8, 10\/'
1o\/§
8.
2
3. 3v'S
3\/5
3.
9. 2\/'2
zfi
9.
4.3f2
4.
3\/'2
10. 8\/'2
sfi
10.
5 . 6/3
6\/§
5.
11 .xx =
= 5500
11.
6.15\/§
6.
15/3
12. Xx =
=5
12.
5
13. x = 2
13. X = 2
14. x = 8
14. X = 8
15- x
= 21
15.
X = 21
x= 5
11
X =
2
17x26
16.
16.
17. X = 6
18. x : 6
18. X = 6
I
I
I
I
\
I
•'
j
273
CHAPTER
ANSWERS TO THE
CHAPTER 30 ANSWERS
THE EXERCISES
EXERCISES
CHAPTER EXERCISE:
EXERCISE:
10. ‑
avoid any
any trickiness,
trickiness, it’s
it's best
7. [fil
E To avoid
best to plug
plug in
numbers . Leta
Let a =
through
numbers.
= 2 and
and b = 2. Going
Going through
each choice,
choice,
each
l.
a
=3
A)
(- 4) 2 = 16
A) (‐4)2
16
1
7
fl : 3
B) (‐4)4
(- 4)4 = 256
B)
C)
16
C) (2-2)2
(2 · 2)2 = 16
D
= 22-·1166 =
D)) 2-24
2 · 24 =
=3322
11=3\/E
= 3y'a
w;
~ = y'a
the largest.
largest.
(B) is the
-91‐a
=a
1
8.
[Q]
of the
the first equation,
equation,
Cube both
both sides
[__D_] Cube
sides
(x2)3 = (y3)3
2-I
x6
P = 233
2x
2.1/
2x‐y=23
2x-y
= 23
= y9
6
9
,
Now
can be
be replaced
replaced by xx6,
N
ow y
y9
can
X32=y9
x3z =y9
x3z=x6
x3z
= x6
x ‐ yy =: 3
xx = yy +
x=
+3
3 2=: 6
3z
z=
2 : 2
3.
[Q]
[E] Raise
Raise each
each side
side to
to the
the 4th
4th power:
power:
9
.‑
9. 0
5 = 10
yy5
= 10
(y5)4
= 104
104
(y5)4 =
Ir'
f!
ti
Vn/Esz-xé
v x = y x • x ==\/x_%=(x%)"l’
y x = (x 2 ) 2 ==xx
y20
3/20 = 10,000
1
2
X \f
31
3
4
3
4.
Therefore,
Therefore , a = 3
4
0
10.
4G-::i,
v x- y ·=(x
2 4 !
y )4=x
2· 1 4.!
! 1
r=
4y 4= x "y =y v x
In
' xbc = x30
xac+bc
___
x30
m+m=w
ac
+ be= 30
(a + b)c=
b)c : 30
5c : 30
Sc=
30
cc== 6
6 [g
.‑
6.
274
I
I
PANDA
THE
COLLEGE PANDA
1HE COLLEGE
11.@]
22(2n+3) =
= 23(n+5)
22(2n+3}
2(2n +
3 (n + 5)
+ 33) )=: 3(n
4n
+ 66 =
: 3n
15
+15
3n +
4n +
n=9
12. 0(
- 2); =
n
=
v'l- 2 . - 2 . - 2 1- - 2 . - 2 = - 2 . 1 - 2 . - 2 =
13.
- 2-¼
[£]
2x+ 3
-
(2x) (23 )
-
2x(23 -
= k(2x)
2x = k (2x)
1) = k (2x)
2x
2x (7)
= k(2x)
7= k
14.
exponents.
the exponents.
Multiply the
[filMultiply
(53)4k
= (5§)24
(5! )24
(53)4k=
= 5s
512k =
Since
can equate
equate
same, we can
the same,
are the
bases are
the bases
Since the
8
2
the
exponents: 12k =
= 8 and
and so
so k =
= % =
= g..
the exponents:
12 3
15.
15.
and
power and
2a power
the 2a
raised to the
means raised
2a means
The 211
[filThe
the
root.
bth root.
the bth
means the
bottom means
the bottom
on the
b on
the b
16. [Q]
@ Multiply
Multiply both
The left
left
together . The
equations together.
both equations
16.
5
hand
The right
side
hand side
right hand
x y5. The
gives x5y5.
side gives
hand side
gives
gives 80.
275
CHAPTER
ANSWERS TO
TO THE
THE EXERCISES
EXERCISES
CHAPTER 30
30 ANSWERS
Chapter 2: Percent
Percent
CHAPTER
CHAPTEREXERCISE:
EXERCISE:
f"o""i:l12.75
‐0.085
_ 8.5
A,
1. ~ 501‐‐
150 =
0.085 =
8.5%
2.
:
2. @32,m0(1.15)
@]3 2,000 (1.15) =
9.
for
9. [D]
~ Let xx be
be the
the sales
sales tax (as
(as a
a decimal
decimal for
now).
at the
now). We'll
We'll convert
convert it to a percent
percent at
the end.
end.
36,800
36,800
105.82(.90)(1+
= 100
105.82
(.90)( 1 + x) =
..O‐5
31%
[]J ~:6‐0.03125~
= 0.03125 ~ 3.1
%
3.
4..
l1 +x=
+x=
.Letz
=1.50(100)
[I]
Let z = 100. Then
Then x
x =
1.50(100) = 150
_
and
is
and y = 1.20(100) := 120. x is
150
- 120
150‐120
120
120
30
30
0
=
%
‘ fi120‘ =z25M
0
Each year,
Veronica keeps
whatever she
she
Each
year, Veronica
keeps whatever
has in her
her account
account plus
the interest
that
has
plus the
interest on
on that
amount. Because
amount.
we can
Because m
misis a
a percentage,
percentage, we
can
convert it to
to a
a decimal
decimal by
convert
100,
by dividing
dividing it by
by 100,
giving
= 11 +
+ 0.0lm.
0.01m.
giving us
us 0.01m.
0.0lm . Therefore,
Therefore, x =
x
X
z 0.05 = 5%
=
xX = 800
Therefore,
Therefore, 940 -‐ 800 = 140 more
more dishes
dishes were
were
served during
served
during dinner.
dinner .
11.
11.
2,690 -‐ 2,
2,690
140
2,140
2,140
x 1100%
0 0 / o~z 25.7%
X
21140
8.
=
(105.82)( .90) -‐1 l
_ (105.32)(.90)
1.175x = 940
l.175x
El
new value
value -‐ old
old va
value
fol new
lue x 100%
_2
O
6-~
Id
I
x
100
1/oold
value
o va
ue
7.
100
X
10. I140 ILet xx be
number of
dishes served
be the
the number
of dishes
served
during lunch.
lunch . Then
during
Then
larger than
larger
than y.
5.
100
100
(105.82) (.90)
(105.82)(.90)
A := ((1.25)
1 2 5( B)
A
7
0 (1.25)
(1.25)(B)
70: =
( B)
6= 8
56
[]2JLet the original
price of the book be $100.
Then James
James bought
Then
bought the
the book
book at
at
100(1‐- 0.20)(1
0.30) =
100(0.80)(0.70)
100(1
0.20) (1 -‐ 0.30)
= 100(0.80)
(0.70) =
=
5
$56, which is ; = 56% of the original price .
1 0
12.
= 24
of chicken
chicken
[I] Kyle
Kyle ate
ate 20(1.20) =
24 pounds
pounds of
wings
= 21
hot dogs
dogs..
wings and
and 15(1.40)
15(1.40) =
21 pounds
pounds of
of hot
That 's a
That’s
= 45
pounds of food.
a total
total of 24
24 + 21
21 =
45 pounds
food.
John
: 35pounds
food.. The
The
John had
had 20+
20 + 15
15 =
35 pounds of
of food
percent
from John
John to Kyle
Kyle is
percent increase
increase from
0
Let x be
be the
of pistachios
at the
the
the number
number of
pistachios at
start.
each day,
what’s left
start. At the
the end
end of each
day, what's
left is
1 -‐ 0.40 = 0.60 of the
the day’s
day's starting
starting amount.
amount .
Over
w o days,
Over ttwo
days,
45
45 -‐35
35
35
z. .29
29 = 2
9°/o
35
~
29%
x(0.60)(0.60)
: 27
27
x(0.60)
(0.60) =
0.36x
: 27
0.36x =
x
X
~
13.
=
= 75
75
Let her
count be
bex.
loss of
of
[I] Let
her starting
starting card
card count
x. A
A loss
18
(0.82)x. From
18 percent
percent reduces
reduces her
her total
total to
to (0.82)x.
From
there,
gets the
there, an
an increase
increase of 36 percent
percent gets
the total
total
to (1.36)(0.82)x.
(1.36)( 0.82)x. Now,
Now,
(1.36)(0.82)x
(1.36)
(0.82)x = n
n
x=~--x : (1.36)
(1.36)(0.82)
(0.82)
276
THE
THE COLLEGE
COLLEGE PANDA
PANDA
14.
[Ij 12,000
(0.94) 10 m
~ 6,460.
6,460 .
12,000(0.94)10
19.
after 55 years
will be
3,000(1.06)5,
but the
the
after
years will
be 3,
000 (1.06 ) 5, but
interest
3,000(1.06)5
000..
interest earned
earned will
will be
be 3,000
(1.06) 5 -‐ 3,
3,000
The total
amount in the
the checking
The
total amount
checking account
account
after 55 years
years will
will be
be 1,
after
1,000(1.01)5,
000(1.01 ) 5 , but
but the
the
interest
1,000(1.01)5
1,000..
interest earned
earned will
will be
be 1,
000 (1.01 ) 5 -‐ 1,000
With
deposit and
With a
a larger
larger initial
initial deposit
and a
a higher
higher
interest
savings account
interest rate,
rate, it's
it's obvious
obvious the
the savings
account
will
difference
will have
have eamed
earned more
more interest.
interest. The
The difference
in earned
will be
(3,000(1.06)5
earned interest
interest will
be (3,
000 (1.06) 5 -‑
3,000)
(1,000(1.01)5
1,000).
3,000 ) -‐ (1,000
(1.01) 5 -‐ 1,000
).
15. I100 ISince
Since scarves
scarves and
and ties make
make up 80% of
the accessories,
accessories, the
the 40 belts
belts must
the
m u s t account
account for
20%. Letting
Letting the
the total
total number
20°/o.
number of accessories
accessories
be x,
x,
be
20°/o of
= 40
20%
of Xx =
1
-5x:40
x = 40
5
xX :=2 200
00
There
are 200 accessories
accessories in the
There are
store.
the store.
Hopefully
you’re able
get this
Hopefully you're
able to get
this without
without
having
make an
an equation,
having to make
equation, but
but there's
there’s no
no
harm
harm in aa little
little algebra!
algebra! Now
N o w we
we can
can
20.
P(1 +- r)5 - P
lOO
and ~g x 200 =
= 120 ties.
the 120
120 ties
ties
and
ties . Half
Half of the
p
(60 ties)
replaced with
scarves, so
so the
the store
store
ties) are
are replaced
with scarves,
will
w
i l l end
+ 60 =
= 100 scarves.
scarves.
end up with
with 40 +
I
16. 1.728 After
years, the
the market
market value
After 3 years,
value of the
the
bond
bond is
9000.2)3
= 900(1.728)
900(1.728) == 900
900(1+
0.728).
900(1.2) 3 =
(1 + 0.728).
Therefore,
Therefore, p =
z .728
[g To get
get the
the final
final value
value after
after aa percent
percent
increase, you
have to multiply
the initial
initial
increase,
you have
multiply the
value
the percentage
(as aa decimal).
value by 1 plus
plus the
percentage (as
decimal) .
So in 2016, Sims
Sims must
spent l.34x
1.34x
must have
have spent
dollars on
dollars
must have
on groceries.
grocerie s . In 2017, she
she must
have
spent (1 +
+1.45)(1.34x)
: (2.45)(1.34x)
spent
l.45 )( 1.34x) =
(2.45)( 1.34x)
dollars
dollars on groceries.
groceries .
18.
[g The
The percent
percent change
change is the
the new
n e w minus
minus the
the
old
old times
times 100. Notice
Notice that
P’s
old over
over the
the old
that the
the P's
cancel
out.
cancel out.
determine
there are
z 40
40 scarves
scarves
determine that
that there
are ~%x
x 200 =
17.
amount in the
the savings
savings account
account
[g The
The total
total amount
|__D_] Let x
x be
be the
the amount
amount of taxes,
taxes, in millions
millions
[Q]
of dollars,
dollars, collected
collected by County
County A in 2016.
Since the
the taxes
taxes decreased
from 2016 to
Since
decreased by 25% from
2017,
(1 -‐ 0.25)x =
: 60
60
0.75x = 60
60
xX =
= 80
80
Because
Bcollected
the same
same amount
amount
Because County
County B
collected the
as County
County A in 2016, County
County B also
also collected
collected
as
80million
dollars of taxes
taxes in 2016. In
In 2017,
80
million dollars
County
collected 20% more
more than
so
County B collected
than in 2016, so
County B must
have collected
80(1.20)
: 96
96
County
must have
collected 80
(1.20) =
million
dollars in 2017.
million dollars
277
X
100
=
CHAPTER 30 ANSWERS TO lliE EXERCISES
Chapter 3: Exponential vs. Linear Growth
CHAPTER EXERCISE:
1
1..
0
home
value of the home
exponential decay. The value
a case of exponential
The situation
question is a
the question
presented in the
situation presented
time .
over time.
increments over
smaller increments
and smaller
smaller and
then by smaller
beginning and
the beginning
decreases
o r e significantly
and then
significantly in the
more
decreases m
decay.
exponential
models
that
graph
a
shows
A
answer
Only
Only answer shows a graph that models exponential
decreases
be stocked
shelves left to be
number of shelves
rate, the number
a constant
shelves at a
constant rate,
stocked decreases
employees stock shelves
the employees
[Q] Since the
2.. lg
function .
linear function.
decreasing linear
Therefore, the function
time. Therefore,
over time.
rate over
constant rate
at a constant
function p is a decreasing
25 20
=
;
be 252‐020
turns out
which turns
increase, which
percent increase,
calculate the percent
need to calculate
we need
growth, we
out to be
exponential growth,
3.. []] With exponential
20(1.25 ) 1,
modeled by P := 20(1.25)',
be modeled
can be
growth can
the exponential
and the
growth factor is 1.25, and
0.25. Therefore,
exponential growth
Therefore, the growth
population.
initial population.
where
20is
is the initial
where 20
4..
increase is 125 -‐
constant increase
[I] The constant
initial
(the initial
y-intercept (the
and the y-intercept
25 and
slope is 25
100 =
: 25. Therefore,
Therefore, the slope
population) is 100.
population)
alloy's
metal alloy’s
20 is the metal
equation, where
definition of f is in the form
form of an
an exponential
exponential equation,
where 20
given definition
[QJThe given
5.. @
second.
each second.
increased each
which itit increased
percent by which
15 is the percent
experiment and
the experiment
beginning of the
at the beginning
temperature at
temperature
and 15is
1
greater
=
+ 1%
growth factor, 1 +
because the growth
second because
each second
increased each
temperature increased
We know
know the temperature
z 1.15,
1.15, is greater
/!
15
= 0.85.
have to be 1 -‐ % =
would have
factor would
growth factor
the growth
correct, the
answer C to be
than 1. For answer
than
be correct,
100
6..
0
f
doubled
infected cells doubled
number of infected
predicted number
the predicted
exponent, the
growth factor is 2 and
Since the growth
and ~5 is the
the exponent,
so oonn .
And so
80(2) 2 := 320. And
C(lO ) =
days , C(10)
10 days,
(2) 1 =: 160. After
days, C(5) =: 80
after 5 days,
So after
days . So
5 days.
every 5
every
80(2)1
After 10
= 80(2)2
7..
II]
8..
increases by 1.002 -‐ 11 =
registered increases
number of cars registered
the number
equation is 1.002, the
the equation
growth factor in the
II] Since the growth
:
the
97% of the
1, 000 (0.97 )/r/( l / 4 ), 97°/o
000 (0.97)4h = 1,000(0.97)"/“/4),
as N =
express asN
can express
we can
which we
model, which
the model,
on the
Based on
Based
: 1,
1,000(0.97)4"
1
1
.
minutes .
every ~3 x 60 =
decrease every
3% decrease
a 3%
That's a
hour. That’s
~ hour.
every 5
after every
remain after
bacteria
= 15
15 minutes.
bacteria remain
1
months). Therefore,
every %year
0.002 =
: 0.2% every
year (6 months).
Therefore, n =
z 0.2
2
reference. The
a reference.
as a
equation y = 1abk
use the equation
decay, we
exponential decay,
a case of exponential
have a
we have
[QJSince we
9.. [E
we can use
11115 as
years .
4 years.
every 4
0.94
=
0.06
1
of
factor
a factor
exponentially by a
decays exponentially
at 14,000 and
starts at
trees starts
population of trees
population
and decays
1‐
every
0.94,k
14000,b := 0.94,k
Therefore, a = 14000,b
Therefore,
t
14,000 (0.94)4
= 14,000(0.94)i
the equation
and the
4, and
z= 4,
equation is P
P=
straight line.
a straight
forming a
closest to forming
the closest
Scatterplot C is the
II] Scatterplot
received
she has
amount she
total amount
Because the total
received: 3, 9, 27, 81. Because
has received:
she has
amount she
total amount
Keep track of the total
11. II] Keep
has received
10.
growth .
exponential growth.
relationship is exponential
each day, the relationship
triples
triples each
decay
linear decay
relationship is linear
decrease, the relationship
constant decrease,
a constant
this is a
Because this
book . Because
a book.
loses a
Albert loses
month, Albert
Each month,
12. []] Each
(decreasing linear).
linear) .
(decreasing
13..
= 80.
c=
count is 80 so
initial count
hour so
every hour
doubles every
count doubles
[QJThe cell count
[El
so the growth
growth factor, r,r, is 2. The initial
so 6
278
THE
PANDA
COLLEGE PANDA
THE COLLEGE
14.
constant.
footage is a constant.
original square
the original
square footage
percent of the
Five percent
CgFive
make it
would make
which would
change, which
doesn't change,
It doesn't
linear
growth..
linear growth
exponential decay.
one of exponential
model is one
year, the model
cut in half
gets cut
items gets
number of items
15. []] Since the number
half every
every year,
equation is the exponential
6, 000. The first equation
500 (4) := 6,000.
1, 500(4)
V = 1,
model: V
linear model:
the linear
equation is the
second equation
16. [[] The
The second
exponential
4
2,800.
=
3,200
6,000
difference
The
3,200.
=
)
(2
model:
V
:
200(24)
=
3,200.
The
difference
is
6,000
~
3,
200
=
2,800.
200
=
:
model
17.
m
start), m
at the start),
= 0 (i.e. at
when t =
50 when
equation . Since P = 50
an exponential
exponential growth
growth equation.
equation is an
given equation
CgThe given
the
So we
equation) . 50
into the equation).
50 into
and P =
plugging in t =
confirm this by plugging
can confirm
m
u s t equal
= 0 and
= 50
we know
know the
(you can
50 (you
equal 50
must
which
each of the two
Now
choice D. N
choice C or choice
answer
o w we have
have to check each
t w o answer
answer choices
choices to see which
either choice
answer is either
equation
the equation
that the
quickly see that
we'll quickly
calculator, we’ll
a calculator,
use a
we use
If we
table. If
the table.
values in the
the values
approximates the
better approximates
one
one better
= 86.12.
with nn =
equation with
the equation
than the
better than
30, 45 better
= 15,30,45
given values
models the given
= 54.38 models
with
values of P for t =
with n =
information,
From the given
reference. From
as a reference.
abt as
equation y := abf
exponential equation
standard exponential
the standard
use the
Let's use
18. [QJ
@ Let's
given information,
equation
exponential equation
growth factor). Since the standard
(the growth
and bb := 1.02 (the
amount) and
initial amount)
(the initial
16 (the
a =
a
= 16
standard exponential
15 5
~
gives k =
which gives
days, which
into days,
hours into
15 hours
convert 15
requires tt and
and k to be
be in the same
same units,
units, we
we have
have to convert
= fl~~ := 5
requires
1
.
W/(SIB) = 16(1.02)
(t ) = 16(1.02)
we get
together, we
everything together,
Putting everything
days . Putting
days.
get g
g(t)
16(1.O2)'/‘5/8)
160.02)?
279
CHAPTER 30 ANSWERS TO THE EXERCISES
Chapter 4: Rates
CHAPTEREXERCISE:
1.
[I!]For
protein intake
intake of 7 x 60
60 =
grams.
For one
one week,
week, Trm's
Tim’s diet
diet plan
plan would
would require
require aa protein
= 420 grams.
Since each
each
Since
7 30
30 = 14
protein
buy 420 +
protein bar
bar provides
provides 30
30 grams
grams of protein,
protein, he
he would
would need
need to buy
14protein
protein bars
bars..
2.
0
3.
[I] The pressure
4.
WThe pool has a capacity
Over 6 years, the screen size increased by a total of 18.5 - 15.5 = 3 inches. That's 3 7 6
each
each year.
year.
= 0.5 inches
increases by 70 - 50 = 20 atm while the submarine descends - 900 - (- 700)
meters.
per meter,
per 10 meters.
meters. That's
That’s 20 7+ 200 =
= 0.1 atm
atrn per
meter, or 1
I atm
a i m per
meters.
=
- 200
of 5 x 300 = 1,500 gallons. At an increased rate of 500 gallons per hour, it
would
would only
only take
take 1,
1,500 7+ 500 =
: 3 hours
hours to
to fill the
the pool.
pool.
5. []]
dollars
20
2051i"'"f''""
a,,nDkrt,
p p l e Xs dd‐ollars
,,nDkrt,
a ~,
6.
""
d II
z?-20d dollars
=
0 ars
a
[f]The racecar burned
22 - 18 = 4 gallons of fuel in 7 - 4
will
will have
have to consume
consume 18
18‐- 66 =
z 12
12more
more gallons.
gallons. That's
That’s
= 3 laps. To get to 6 gallons left, the racecar
lapss
43
3 laps
12gallon§x
‐ 9 9 more
more laps
12
_galk,nsx _galk,ns=
laps
4
which is Lap
Lap 7 +
+9=
= 16.
which
7. I100 IIt took 2.5 hours for 65 - 40 = 25 boxes to be unloaded. There are 3.5 hours from 3:30PM to 7:00PM.
25 boxes
.
hours, 3.5.het:rrs
3 . 5 h o u r § x M .het:rrs =
: 35
35 more
more boxes
boxes will
will be
unloaded. That’s
total of 65
65 +
+ 35
35 = 100
In 3.5 hours,
be unloaded.
That's a
a total
2.5mm
25
boxes.
boxes.
Average speed
speed is just
just total
total distance
distance over
over total
total time.
The total
was 2400 x 12
12 =
:
8. I120 IAverage
time. The
total distance,
distance, in inches,
inches, was
28,800.. The
The total
seconds, was
w a s 4 x 60 =
: 240. 28,800
second.
28,800
total time,
time, in seconds,
28,800 7+ 240 z= 120 inches
inches per
per second.
432 I
9. 1432
.
90
90 words
words
=
l m e u t é § xx fi
‐ 432
.
12.mi:m:rtes
432 words
words
2 .5 .mi:m:rtes
:_w- --~ m
l ~m,SJOn
180
100 ~
1 jar
xX ‐OO‐‐‐1’rl
60 jars
l :_ s- --~ m xX _2O10_ _m) amr ==60jars
15 ..!c!
.l-l...l-'H'tTJll:,SJOn
2
;n
nr~
~ ......s
11. [[]
6 0.mim:rtes
m m ? 32
32 kilometers
kilometers
.,
60
~ 265
kilometers
22m
265 kilometers
.het:rrs xX f l.lwtrf
xX _ .mi:fttrtes ~
14 5
m ‐ s
1n7
3 liters
8 dollars
11 hours
hours x 3hliters x
xm I'
: 132 dollars
dollars
12..@
~ 11
=
1 liter
1ter
2 hours
ours
280
THE COLLEGE
COLLEGE PANDA
PANDA
3
@3cu
s
r o i l xX
13. [Qj
3~
14.
-é cups
cups of lye
lye
2 ‐ 2 ‐=
2
= 11.25
11.25 cups
lye
cups of
of lye
2
gw
5~
An 88 inch
inch by
10 inch
inch piece
piece of
of cardboard
cardboard has
has an
10 =
z 80
inches.. A
A 16
16inch
inch by
by 20
20 inch
inch
[I]An
by 10
an area
area of
of 8
8 x 10
80 square
square inches
piece of cardboard
cardboard has
has an
an area
area of 16
16 x 20
20 =
= 320 square
piece
square inches.
inches.
£ ~
o.
nx
320
l1ci
15. ~
-1,000kulacl€x
15.
1,000.hmrct< x
16.
17.
2 dollars
‐2d‐°"a.rS‐‐
=8dollars
~=
0
W 8 dollars
80
8
29 ¢k6f
'
29
2 large bahar
fl
.kalaci<
xx M _Fiko[ ~
16 large bahar
400
400144th
9M z16largebahar
9
0
The first
first 150 miles
miles took
took 150 -;-30
+ 30 == 55 hours
hours.. The
next 200 miles
took 200-;200 + 50
: 44 hours
hours.. His
His average
The
The next
miles took
50 =
average
speed, total
total distance
distance over
time, was
was (150 +
+ 200) / (5 +
+ 4) m
miles per
speed,
over total
total time,
~ 38.89 miles
per hour.
hour .
The clock
clock falls behind
behind by 8 minutes
minutes every
every hour.
hour. There
are 6.5 hours
hours between
between 4:00 AM and
10:30 AM
A M,
CgThe
There are
and 10:30
so the
the clock
clock falls behind
behind by 8 x 6.5 =
: 52
52 minutes.
minutes. The
The correct
then 52
52 minutes
minutes past
past 10:30
10:30 A
M,
so
correct time
time is then
AM,
which
which is 11:22 AM
AM..
18.
0
2240
Jared’s rate
rate is E~ =
= 16
16 pages
pages per
per hour
hour.. Robert's
Robert’s rate
then be
be 16
16 x 2 =
= 32
32 pages
pages per
Jared's
rate must
must then
per hour.
hour . It
1
would take
Robert %
hours to review
review the
report. That’s
= 225 minutes
minutes..
would
take Robert
~; = 3.75 hours
the 120-page
120-page report.
That's 3.75 x 60
60 =
15 h
7.1 x 10
1015
’ ns
'A77.1
~
19
19.. ~
A
20.
1
JHL
lm
lf
0.8 grams
grams
0.8
4.8 X 1023 ~ X
x4.8><1023hy,dmgerr‘iofi§x
X
1000m‘lf ~
_ grams
1000.mt"
z 1.2x
‑ 1L
2 10-51‘3‐ra‐’ms
0 5 - L1L ~ 1. X l
L
What makes
makes this
this question
question aa little
little tricky
tricky is
is that
that we
we don't
distance Brett
[I] What
don't know
know the
the distance
Brett travels
travels each
each month
month
the number
gallons he
he uses
uses each
each month
month.. But
let’s say
he needs
or the
number of gallons
But we need
need to start
start somewhere,
somewhere, so
so let's
say he
needs 2
gallons of gas
gas each
each month
(you can
can make
make up
up any
any number
want).. That
means he
60
gallons
month (you
number you
you want)
That means
he travels
travels 30
30 x 2 = 60
miles
each month
dollars (ridiculous,
(ridiculous, I1 know)
miles each
month and
and each
each gallon
gallon costs
costs 160 -;-2
+ 2 =
= 80 dollars
know).. Now
N o w if he
he switches
switches
the new
n e w car, he'll
he’ll only
only need
need 60 -;-40
+ 40 =
= 1.5 gallons
gallons of gas
miles divided
divided
to the
gas each
each month
month (distance
(distance of 60 miles
by the
40 miles
miles per
per gallon).
gallon). Because
the price
price of gas
stays the
i l l cost
cost him
him
the 40
Because the
gas stays
the same,
same, that
that amount
amount of gas
gas w
will
80 =
z 120 dollars
dollars each
each month.
month. The
The answer
answer ends
what number
number we
we make
make up
up
ends up
up being
being 120 no
no matter
matter what
1.5 x 80
month.
for the
the number
number of gallons
gallons of gas
gas Brett
Brett uses
uses each
each month.
.
160
Here's
pnce per
per gallon
gallon of gas,
gas, then
Brett currently
currently uses
uses -7
Here’s an
an alternative
alternative solution
solution.. If we
we let
let x be
be the
the price
then Brett
X
160 ((30)
4,800
gallons
means that
that he
he drives
30 ) = 4’
--800 miles
m iles each
eac h
gallons of gas
gas each
each month
month.. Using
Using that
that amount
amount of gas
gas means
drives -152‐0
X
X
12
44, 800
120
month.
he will
gas each
each month
that distance.
distance.
month. With
With the
the new
n e w car, he
w i l l need
need ' BOO-;‐Z‐ 40
40 = 70 gallons
gallons of gas
month to drive
drive that
x
X
X
120
Since each
each gallon
gallon of gas
gas costs
costs x
atdollars,
he will
will need
: 120 dollars
Since
dollars, he
need to spend
spend -% x x =
dollars on
on gas
gas each
each month.
month .
X
Each jar of honey
costs 9 -;-4
+4=
= 2.25 dollars
dollars.. She
15+
dollars. That's
That’s a
21. I48 j Each
honey costs
She can
can sell
sell each
each jar for 15
-;-33 = 5 dollars.
a profit
profit
: 2.75 dollars
dollars per
per jar. To make
make aa profit
profit of 132 dollars,
would have
have to sell
sell 132 -;-2.75
+ 2.75 =
= 48
48
of 5 -‐ 2.25 =
dollars, she
she would
jars.
jars .
281
281
ANSWERS TO THE EXERCISES
CHAPTER 30 ANSWERS
CHAPTER
Chapter 5: Ratio & Proportion
EXERCISE:
CHAPTER EXERCISE:
6.@
6. @J
value of b is
1. ~ Since a z= 28, the value
g
then
value of c is then
; x 28 = 24. The value
5
5x24‐15.
BX24 = 15.
2.
1
=
Vnew =
Vm’w
essentially just
are essentially
ratios are
that ratios
Remember that
[lJRemember
2
nr h
3mzh
Vold =
void
2
1
3
2
7r(0.80r)2(1.10h)
0.80r) (1.10h)
n(
(~m
2h) = 0.704V
= (0.80)2(l.10)
0.704v0m
(0.80)2(1.10) (értrzh)
0 1d
equivalent to
ratio is equivalent
given ratio
So the given
fractions. 50
fractions.
zl+1l‐_+ = -4 XX -3 =‐ §3-2‐=33:. 22
119392 9
2- ...!... } - = - ...!... 4·2
2 _4
4·
4
decreases by
volume of the cone decreases
The volume
1 ‐- 0.704 := 0.296 = 29.6%.
Therefore, n =
Therefore,
= 3.
7. [[]
the
Y. Then
Product Y.
price of Product
be the price
@J Let y be
3. [El
Then the
price of
the price
and the
Xisis l.25y
Product X
price of Product
price
1.25y and
ratio of
the ratio
Simplifying the
0.75y. Simplifying
Product Z is 0.75y.
Product
we get
prices, we
their prices,
their
Anld =
1
§(bl + b2)h
l
Anmu : E
= 1.25 = 5 : 3 = 5 X 4 _ 5 _ 5 :33
l.25y
1-25.v_@_§-§_§xe_§_5
33 ‐ 33 ‑
4
4 ‐4
0.75 4 ' 4
0.75y
0.75y‘0.75‘4
(Ebl +
1
l
2172) (2h)
1
1
= (§)(2> [5071+ w]
1
= 5071+ bzlh = Aold
4.@]
4.
@
2
vv2
Pold
R
Paid== F
P
P
1
.
b1 and
Notice
u t from
and
from In
factored oout
was factored
how %was
Notice how
2
51)
- 0o225P
o.25v 2 _
- 0.25v2
(o.sv )2 _
-_ (0.SV)2
new -‐
R
‐-
R
-‐
-·
b.
stays the same.
same.
area stays
b2. The area
old
and
Kevin's sphere,
radius of Kevin’s
be the radius
8. [[] Let r be
sphere, and
Calvin's
radius of Calvin’s
be the factor the radius
let x be
greater by.
sphere
sphere is greater
what
fourth of what
drops to a fourth
power drops
The electric power
it was.
was.
2, where
where ss is
square is A =
area of aa square
5. [[] The area
= s52,
the length
each side.
side.
length of each
.The
VKc’vin :
A
m , = (1.105)2
: 1.21s
1.2152
(1.10)2s2 =
(l.10 s) 2 = (1.10)2s2
A new=
4
57"
2
4
=
l.21A old
= 1.2mm,
Vca lvin
greater .
21% greater.
area is 21°/o
new area
The new
= 43n(xr) 3 = x 3 ( 37Tr3 ) = X 3 V Krvin
=4
X = v'4~ 1.59
x3
282
THE
COLLEGE PANDA
THE COLLEGE
PANDA
9.
9. @
E The
The area
area of the
the original
original triangle
triangle is
is
11
_ 11 22
-§(5)(5)
(s)(s) =
_ -s
2
225
Amy
An
ew
1
13.
14.
1
1 ( XS )2 = Xx2
2((552)
1 2) xZAOM
2
§(xs)2
=2
2s = X Aold
~M
l~ :a~erback
ar cover
4 hardcover
paperback
copies
paperback copies
[QJ
@
)!_
= 2.7
1__EZ
2.4 _ 3.6
2.7 ( )
2.7
y = 3.6 2.4
xx2=0.64
= 0.64
2
=
x = . 8.80
0
X
s5 must
decreased by
must have
have been
been decreased
11 -‐ .80 = 0.20 = 20%.
15.
99
= 1.8 = 5
II] The
The ratio
the weight
weight of Box A to
ratio of the
to the
the
weight of Box B reduces
7: 5. Since
weight
reduces to 7:5.
Since the
the
weights of Boxes
Boxes C
C and
follow the
the same
weights
and D follow
same
E]
10. [Q]
7
7
ratio, Box C must
‐ fi; of
Ofthe
the
ratio,
must weigh
weigh m:
=
2
7 5
total
total weight
weight.: Therefore,
Therefore, Box C weighs
weighs
Lotherslar : 47rd2b
Lstar
(2b)
Lstar =
Z 4rr(3d)2
47T(3d)2(2b)
7
2
= (3)2(2)(4rrd
b)
(3)2(2)(47Td2b)
% x 180 = 105 pounds.
pounds.
12
=
otherstar
: lBL
18Lother
star
11.
x 50
hardcover =
50hardcover
= 125
II] Let
Let x be
be the
the fraction
fraction that
that Star
Star A's
A’s distance
distance
is of Star
Star B's.
B’s.
1
LstarA
Lsmm =
= 5 LstarB
LSmrB
9
2
47r(xd)2b =
: ~(4rrd
%(47rd2b)
4n(xd)2b
b)
2
2
: i(4rrd
game»)
xx2(47td2b)
(4rrd2 b) =
b)
X
x
2 z _
11
=-
9
1
xx =-1
“ 33
The given
given ratio
ratio means
means that
that there
there are
are
12. I20 or 65 IThe
17
eighth graders
17 sixth
sixth graders
graders and
and 28
28eighth
graders for
every batch
batch of 17
17+
28 =
= 45
45 sixth
sixth and
and eighth
eighth
+ 28
every
This means
means that
the total
total number
number of
graders. This
graders.
that the
sixth and
and eighth
graders must
must be
bea
multiple
sixth
eighth graders
a multiple
the case
case that
that this
this total
total is
is 45,
45, the
the
of 45. In the
remaining 110 -‐ 45
45 = 65
65 students
students must
must be
be
remaining
seventh graders
graders.. In the
the case
case that
that this
this total
total is
is
seventh
45 x 22 = 90, the
the remaining
remaining 110 -‐ 90
90 = 20
20
45
students must
m u s t be
be seventh
seventh graders.
graders. Therefore,
Therefore,
students
the possible
possible values
values of nn are
are 20
20 and
and 65. Notice
Notice
the
that we
we don’t
have to
to consider
multiples of 45
45
that
don't have
consider multiples
higher than
since those
those multiples
multiples would
would
higher
than 90
90 since
exceed the
the total
total number
number of students.
students.
exceed
283
CHAPTER 30 ANSWERS TO THE EXERCISES
Chapter6: Expressions
CHAPTER EXERCISE:
0 We factor
factor out
out 6xy
6xy from
from both
both terms
terms to get
get 6xy(x
6xy(x +
+ y).
y).
IDl
.
.
1l
3
4
3a
4 + 3a
2.
The least
2. ~
.The
least common
common denominator
denominator 1s
is 4a. So, -‐ +
+ -g = -i +
+ - = -4 + 3a
a 4
4a 4a
4a
1.
1.
3_a
3..
2
2
2
Expanding, (x
(x22 + y)(y
y)(y + z) = xxzy
[fil Expanding,
y + xx22
z + yy2
+ yz
4..
[g
Divide
Divide the
the top
top and
and bottom
bottom by 4 to get
get
11 +21:
way to get
same answer
the
. Another
Another way
get the
the same
answer is to split
split the
3x
;}x.
fractions
fractions and
and reduce.
reduce.
5..@3x4
@]3x 4 -‐ 33 = 3(x
1)(x 2 -‐ 1)
l)(x ‐- 1)
3(x44 -‐ 1)
1) = 3(x
3(x22 +
+1)(x2
1) =
: 3(x
3(x22 + l)(x
1)(x +
+1)(x
.The
6.. [IjThe expression
b2 pattern,
+ 1.
expression follows
follows the
the (a+
(a + b)
b)22 = a1122 + 2ab
Zab + b2
pattern, where
where a = x + 1
1 and
and b = y +
1. Therefore,
Therefore,
the
expression is equivalent
+ 2)2.
the expression
equivalent to ((x
((x + 1) + (y + 1))2 =
= (x + y +
7
'Elxy‐x
[Q]xy - x2Z= Jx (yW
- x) ‐= X
- x (L
x -M
y) = _x
~
·
xy-y
x y ‐ y22 _ yy(x
( x -y)
‐y)
y(x-y)
y(x‐y)
yy
+ 5 _ 33(x
( x ‐- 11)) ++22(x
( x ++55))
rr7
.
.
.
.
x-1 1 x
x+S
8.. L.::JAdding
Adding the
the two
t w o fractions
fractions in the
the denommator,
denominator, -x g - +
+ - 3- =
6
2
6
3
_5x+7
=
-5x 66+- 7 .
6
‘ result
fl'1p1
't: Sx
‐. ~
Now,
this
it:
Now, 11 over
over thls
result means
means we
we can
can flip
5x
+ 7.
7
2+ 1
2 + -x
2x + 11
2x
-x + -x _
9 · CE]
--f
=
2x
f=
'
1 -2_x_1
2- - x
X
10.
x
X
x
X
2x + 11
2x
-x- x
x
2x
_ 2x
2x +
+1 x
2x +
+1
2x
1
=
-xx
2x
1
=
2x
1
2x‐1
x
2x‐1
2x‐1
x
X
2
[g First
term s. Then
First factor out
o u t an
an 8 from
from both
both terms.
Then use
use the formula
formula aa2‐
b2 = (a -‐ b)(a
b)(a +
b2
+ b).
1
1
1
Bx2 - 1y 2 = 8 ( x 2 - 1 y 2) = 8 ( x - 1 y ) ( x
4
8x2 “ 2
53/2 2 802 ‘ E312)
16
1
1y )
:8("‘1M”+ 411>
Therefore, c =
Therefore,
1
4.
2 (x 2 - 4) + = x 4 - 4x 2 + 4. Now we can apply the formula
2
11. ~ First,
(x + 2)(x
First, expand:
expand: xx2(x
2)(x -‐ 2) + 4 =
z xx2(x2
‐
4 z x4 ‐ 41x2
N o w we can apply the formula
2 - 2) 2 .
2
2
aa -‐2ab+l72
2ab + b = (a
- b)2, where a = xx2and
b=
2, to get x 4 -‐4x2
4x 2 +
(a‐b)2,wherea
andb
=2,togetx4
+44 = (x
(x2‐2)2.
12.
2 ) + (- 4x [fil Combining
terms, we
Combining like
like terms,
we get
get 3x
3x33 + (8x
(8x22 + 7x
7x2)
+ (‐4x ‐
13.
.Combining
like terms,
terms, Sa
5a-‐ 2a
2a=
3aand
and 3y'a
3\/_ -‐ 5/a
5\/E =
= -‐2\/E.
[g
Combining like
= 3a
2/a .
137
w
11x)
11x) ‐- 7 = 3x
3x33 + 15x
15x22 -‐ 15x -‐ 7.
7.
_1+
9(2y)2 +
2(6y) 2 __ 36y2
ny 2 _ 1 1_§
_ 3
+2(6y)2
36y2 +
+72y2
1
14· 3 9(2y)2
2
2
8(3y)
72y
-‐
-‐ 2 + 2
8(3y)2
72y2
_ 2
15.
II] To get
get aa common
c o m m o n denominator
denominator of 2(x
2(x -‐
2) for both
fractions, we
both fractions,
we first factor
factor oout
u t aa negative
negative from the
the
second
bottom of the
fraction by 2:
second fraction
fraction and
and then
then we multiply
multiply the
the top
top and
and bottom
the first fraction
x +
x
= Xx +
x
:
x _
x
=
2x
_
xX
=
xX
X
X
X
X
X
2X
-m
x ---2 + -22(_(22
___
x
_)
=
x
---2
+
---2-(
x---2-)
=
x
---2
2
(x
-2)
=
2
(
_
x
___
2
_)
2
(x
-_
2
)
=
2
(
.
,......
x--‐x)
x ‐ 2
‐2(x‐2)
x ‐ 2 2(x‐2)
2 ( x ‐ 2 ) 2 ( x ‐ 2 ) 2 ( x ‐ 22~)
)
284
THE COLLEGE
COLLEGE PANDA
THE
PANDA
Chapter 7: Constructing Models
CHAPTER EXERCISE:
1.
1.
0
In the
the 2y
the first
first y hours,
hours, the
the carpenter
carpenter lays
lays (x)(y)
(x)(y) =
= xy bricks.
bricks. In the
2}; hours
hours thereafter,
thereafter, he
he lays
lays
(g)
(Zy) == xy
xy bricks.
bricks. Altogether,
Altogether, that's
that’s xy + xy =
= 2xy bricks.
( ~) (2y)
bricks.
2..
For d dollars
dollars worth
worth of mozzarella
mozzarella to have
have been
been sold,
861‐75 pounds
must have
been sold.
sold. That
That leaves
leaves
[I] For
sold, _!!__
pounds must
have been
8.75
d
pounds still
still available
available for sale.
sale.
175 -‐ ‐ _~ pounds
88.75
5
The store's
store’s monthly
monthly total
total cost is
is 3,000
3,000 + 2,
an entire
entire year,
year, we
we multiply
multiply by
by 12
12 months:
months:
3.. @ The
2, 500x. For an
c=
=12(3,000
+ 2,500x
2,500x).
C
12(3,000 +
).
4.. @ The
each customer.
customer. The
cost for all
all the
the customers
The setup
setup fees amount
amount to 100c,
100C, $100 for each
The monthly
monthly cost
customers
amounts
months, the
add up to 50c
amounts to 50c,
50C, $50 for each
each customer
customer.. Over
Over m months,
the monthly
monthly charges
charges add
50C x m, or
50cm. The
charge is therefore
The total
total charge
therefore 100c
100C +
+ 50cm.
50cm.
5.. @ For
2mn. Since there
there are
each pizza,
For mn
mn students,
students, the total
total number
number of slices
slices must
must be
be 2mn.
are 8
8 slices in each
pizza, the
the
2mn
mn .
school
school must
m u s t order
order -T - = T pizzas
pizzas..
8
6..
4
.
d
.
.
[Q]
!!_degrees
degrees per
minute.
So after
[ElThe compound's
compound’s temperature
temperature increases
i n c r e a s e s by 5
per m
i n u t e . 50
after x minutes,
nunutes, the
the temperature
temperature
m
d
dx Th final
.
.· th
dx
._
b y -Ex,
d
dx
dx .
increases
x, or -.a.
e final temperature
temperature 1s
en tt +
i n c r e a s e s by
The
15
then
+ -E'
m
m
m
7.. [Q]
Over 44 days,
days, they
a total
IE The
The bakers
bakers make
make 3xy cupcakes
cupcakes each
each day.
day. Over
they will
w i l l make
make a
total of 44 x 3xy =
z 12xy
cupcakes.
the total
total number
divided by the
number of
cupcakes. The
The number
number of boxes
boxes needed
needed is the
number of cupcakes
cupcakes divided
the number
12
= 12y.
xy =
cupcakes
cupcakes that
that can
can fit
fit in each
each box:
box: icy
X
8.. @ The reduced
So the
the first souvenir
souvenir costs
and the
the
reduced price
price of each
each souvenir
souvenir after
after the
the first is 0.6a. So
costs a dollars
dollars and
remaining
n
l
souvenirs
each
cost
0.6a
dollars.
Therefore,
the
total
cost
is
a
+
(n
l)(0.6a
).
remaining ‐ 1 souvenirs each cost 0.611 dollars. Therefore, the total
‐ 1)(0.6a).
9..
[fJDuring
should go
up and
and be
relatively steep
During the
the biking
biking portion
portion of the
the commute,
commute, the
the graph
graph should
goup
berelatively
steep since
since Kaiba
Kaiba
minutes ,
covers the
walks . When
the rest
area for 15
covers
the initial
initial 4 miles
miles at
at a faster
faster pace
pace than
than he
he walks.
When Kaiba
Kaiba stops
stops at
at the
rest area
15minutes,
the graph
ld be
distance during
time. After
the rest
the
graph shou
should
be flat since
since he
he does
does not
n o t cover
cover any
any distance
during this
this time.
After Kaiba
Kaiba leaves
leaves the
rest
area,
incline since
he covers
commute at
area, the graph
graph should
should go
go up
up and
and be
be at
at aa gradual
gradual incline
since he
covers the
the last
last mile
mile of his
his commute
at aa
walking
criteria.
walking pace.
pace. Only
Only the
the graph
graph in answer
answer C fulfills all of the above
above criteria.
10..
[Q]
his commute,
commute, we're
that goes
[E]Since Mike's
Mike's distance
distance from
from home
home increases
increases during
during his
we're looking
looking for a graph
graph that
goes
up and
slowly at first and
and then
quickly later,
later,
and to the
the right.
right. Since his
his distance
distance from
from home
home increases
increases slowly
then more
more quickly
we're
slope (more
(more steep).
steep). Only
we’re also
also looking
looking for the graph
graph to
to go
go from
from aa low
low slope
slope (less steep)
steep) to aa high
high slope
Only the
the
graph in answer
answer D meets
meets these
these conditions.
conditions.
graph
11..
d
[f] Since
the cost per
per game
dollars .
Since p tokens
tokens can
can be
be used
used to play
play E_
5 games,
games, the
game is _!!.__p=
= dw dollars.
-E
w
w
285
£1710
p
CHAPTER 30 ANSWERS TO THE EXERCISES
rr7
24,500
24,
500 -‐ 17,900
17,900
6,
6,600
plane descends
The plane
=
16200 := 550
descends at a rate
rate of T
=
550 feet
feet per
per minute.
minute . Since
Since the
the plane
plane
12
12
started its descent
started
A =
=
descent at an
an altitude
altitude of
of 24,500 feet,
feet, its
its altitude
altitude after
after tt minutes
minutes can
can be
be represented
represented by
by A
24,500
24,
500 ‐- 550t.
12.
~
13.
[fil The
passenger spent
The passenger
spent 24
24 ‐-
14.
[Q]Mark’s
Mark's annual
salary does
does not
[El
annual salary
l s t , when
not change
change during
during the
the year
year except
except for
for June
June 1st,
when itit ”jumps”
"jumps " by
by $15,000.
$15,000.
a
a dollars
dollars on
on additional
additional miles
miles after
after the
the first,
first, which
which means
means the
the passenger
passenger
24 -‐ aa
24
traveled - b- additional
traveled
additional miles
Adding the
miles after
after the
the first.
first. Adding
the first
first mile
mile then
then gives
gives us
us the
the total
total distance
distance
b
traveled :
traveled:
24
2 4-‐ a + 1 _ 24
2 4-‐ aa +é_
b
244-‐ a + bb
2
b‑
- b
b -+ l =b
-+
b
=
b
_
b
b
Therefore, the correct
correct graph
graph should
Therefore,
jump at
at the
of each
should be
be flat throughout
throughout each
each year
year except
except for
for aa jump
the end
end of
each year
year
we are
(since we
are starting
starting from June
June 1st,
l s t , the
the end
each year refers
the
next
June
lst).
Only
the
end of each
refers to
to the next June 1st). Only the graph
graph in
in
answer D fits this description.
answer
description .
15. @
[Q]Initially,
members are
are each
each responsible
Initially, the members
responsible for paying
paying 5
.!_ dollars.
dollars . But ifif k
k members
members fail
fail to
to pay,
pay, then
then
m
r
m
k
members
remain and
each one
one becomes
m ‐ k members remain
and each
becomes responsible
responsible for _!__ k dollars.
This amount
dollars. This
amount is
is greater
greater than
than
m -‐ k
the original
original amount
amount by
the
mr
mr r- ‐ m
r
r _
mr
_ rr(m
( m-‐ kk ) _ m
kr
dollars
mrr + kkrr _
(( m-‐ kk)) dollars
m(m
mm
m -‐ k ‐- E =
‐ m
( m-‐ kk)) m
( m-‐ kk)) ‐
m
_ m
m (m
mm
(( m-‐ kk)) =
m
. ..
.
rr
Here is an
an alternative
alternative solution.
solution . Since each
each member
member is initially
initiall y responsnble
Here
dollars, the club
loses
responsible for -a dollars,
club loses
m
out
on L
!_ (k) dollars
dollars when
wh en the
the k members
members fail
fail to pay. To make
o
u t on
make up this lost
lost amount,
amount, each
each of
of the
the remaining
remaining
m
m ‐- k members
members must
must pay
m
pay an
an additional
additional
r
- (k )
kr
kr
m
z _n
m
ll
.!!1__
=
_i - =: ---dollars
m -‐ k
m -‐ k
m (m
m
m
m
( m-‐ kk)) d o ars
286
286
THE COLLEGE
PANDA
THE
COLLEGE PANDA
Chapter 8: Manipulating & Solving Equations
EXERCISE
EXERCISE1:
1. rr =
=
1.
2. r
19.
19.
± ~ fl7r
X
X
_ CE7T
=
_ 271
2
_ YY +
+Z
Z
X ( Y++Z)Z =) : X
X ++ ll
X(Y
X Y++XZ
X Z- ‐XX=: 1
XY
2A
2A
3. b =
z h
‑
3.
b
h
X ( Y++ZZ-‐ 11)) == 1
X(Y
V
= lh
4. w
5. hI =
z -‑
5.
I
1
x-111
:
‐
Y+Z‐1
X
V
~
2
m2
?Tr
6. rr =±
z j:
6.
20.
20.
~ 17th
= ±i J c2e2 -‐
sz W
s=
W
7.
7. bb :
8.
_ xX++ 1l
=
x(y + 2) = y
xy + 2x = y
2x = y - xy
2x = y(l - x)
a1122
_ 27rr2
9.
_ S
9. h
h=
S - 2m2
27rr
2rrr
bc
be
10. aa =
‐ ti
7
10.
be
21. First,
First, cross‐multiply.
cross-multiply.
11. d =
= -E
11.
a
m:=
12. m
2ac=ab+b
2ae
= ab + b
2 a c- ‐ ab
a b=z b
2ae
a ( 2 -c ‐b)b )=: b
a(2e
y- b
“ll‐b
-x
x
13. 3/2
: m(x2 -‐ x1) +
+ Y1
y1 =
= mx2
mxz -‐ mx,
m x 1+
+ yY1
1
y2 =
_ mx2 - y2 + y,
14.. X1
x 1- :
" y z +3“
14
m
‐ m e
15. a =
15.
18. p
‐
b
2c‐ b
v2
u2
02 -‐ 142
25
25
I~I
3t
22.
±/f;bx
22.
47r2L
4n 2 L
23. Divide
Divide both
both sides
sides by 33 to
to get
get xx +
+ 2};
= g
23.
2y =
16. y =
_ :i:
16.
17.
z
17. g
g=
a‐
a-
7
[i]
72
t‐2
t2
A2
A2
7t2r2
= 22
7T r
24. Multiply
Multiply both
both sides
sides by
get 2x
2x + 10
10=
by 2 to
to get
= j 4b
- q
2t = %,
a25. Since 2t
a
1
, we
we can
can multiply
multiply both
both sides
sides
M
by2toget4t=
@
by 2 to get 4t =
287
I
CHAPTER 30 ANSWERS TO THE EXERCISES
26. Cross
Cross multiply.
multiply.
31. n
3(p
- h) = 2(p + h)
3(p‐h>=2(p+h)
1
=
x
3
h ==22p
p ++22h
h
3pp-‐ 33h
p=
= 5Shh
Jx + 1
( Sx2
- 3-
3)
32. a = 5(c+ l) 3-c
b2 + 2
5-‑
t=~
P_
33.
Cross multiply.
27. Cross
multiply .
2
k(x2 +
+ 44)) + kkyy =
2
( 1++2r)
2 r=
) =11- ‐ tt
2(1
2 + 4r
4 r== 11- ‐ tt
, = 1-1- 4,
x
2+
7x24‐3
7x
2 3
2
7x2 + 3
2 + 4 + y) =
k(x2+4+y)=7x
k(x
+3
2
2
k_
7x2+3
k=
7x
+3
2
_ 2(x2+4+y)
2(x + 4 + y)
1
28. Square both sides to get (xY)2 = x 2Y = [{]
34.
29. p:
p=
(x3 - x2)(x5 - .x4)
2
30. m =
a
3 a++xx++33 =
=b
axx++ 3a
a
( x++3)
3 )++ (x
( x++ 3)
3 )=
= bb
a(x
((x+
x + 33)(a
) ( a+ 1l )=
) =b
x(X3-- 1)+- 12
X
x2
x
+1
b
x + 3 = -a ++ l1
x n-31
b
:
x =I
b l ‐3
a+
288
288
THE
PANDA
THE COLLEGE
COLLEGE PANDA
CHAPTER EXERCISE:
EXERCISE:
CHAPTER
1.@]
7.
7.
(a + b)3
2.
= (- 2) = - 8
[El The
The answer
obvious just
just by
@]
answer should
should be
be obvious
Cross
multiply to
get 12
12=
+ 2x. Then,
Cross multiply
to get
= kx +
Then,
12
12 -‐ 2x
2x
k:
x
8.
8. E
~ Note
Note that
that
lookingat
Testing n
gives us
us::
looking
at it. Testing
n=0
0 gives
2
0
k=---
3
(- 6)2 =
(‐6)2
= 36
36
2
(0
( 0- ‐ 4)
4)2 = ((00++ 4)
4)2
which
when xx = -‐ 33.. Then
Then
which happens
happens when
2
9.
xx2=
= ((‐3)2
- 3)2 = 9.
(- 4)2 =
(4)2
<‐4)2
: (4)2
9
9.0. ‑
16 == 16
r-
We could
could also
and solve
We
also expand
expand and
solve like
like so:
50:
_ p(( 1(1++i ) " ‐ 11)
J=
f‘
(
i
)
fJi
1=
( ( 1 + 1i)"
) "-‐ 11))
=pp((l
Ji
fi
_
(1 + i)"
m
i l- 1‐ =Pp
(n
4)(nn -‐ 44)) = (n
+4)(n
(n -‐ 4)(
(n +
4)( n + 44))
n2‐8n+16=n2+8n+16
n2 - 8n + 16 = n 2 + 8n + 16
6 n=: 0
-‐ 116n
n=
=O
0
3.‑
10.
3. []]
271?
2m '
= aacc
b=
11.
If
: ac, then
If bb =
then b -‐ ac
ac must
must equal
equal 0.
@If
" _
- · -=E'
2
l · -=-
m
m _2
2 Z4
_
8
8
.x2
5x -‐ 24
be factored
factored as
as
[fil
x 2 + 5x
24 =
= 00 can
can be
(x +
The two
t w o possible
possible
+ 8)(x
8)( x ‐- 3) = 0. The
solutions are
are then
then -‐88 and
Since k < 0,
solutions
and 3. Since
k=
= -‐ 88aand
n d lkl
|k| =
= 88..
4
3 , then
3x =
: -‐15.
4.. [Q]u 3
3xx -‐ 8 z
= -‐ 223,
then 3x
15.
Multiplying both
0 and
and
Multiplying
both sides
sides by
by 2,6x
2, 6x = -‐ 330
6x
‐
7
z
‐37.
6x = - 37.
m
·w
12
Cross multiply.
multiply.
Cross
-4 = -38 m
Gf = %2
9
12 = 72m
y3 : J_1
1
8
32
8
2
8
yy 2==32‑
1
6
-= m
6.0
3x
n
n
l
which
means -1 =
= -1.. Then,
which means
Then,
m
m
4
n
l n
l 1
1
ac
0
Multiply both
sides by
get %
4,
Multiply
both sides
by 22 to
to get
m = 4,
.
2 := 1l
~
5.
5.
0
+1= -
y=ti =1
8
3x
=-9
X
=-3
(x+2)3
: (‐3+2)3
z -‐11
(x
+ 2) 3 =
(- 3 + 2) 3 := (‐1)3
(- 1)3 =
289
CHAPTER 30 ANSWERS
CHAPTER
THE EXERCISES
ANSWERS TO THE
EXERCISES
13.‑
13.1771
17. I30 ITo make
make this
this problem
problem easier
easier to
to work
work
2\/x
2y'x+4
3
i.
with,
let A =
Then,
with, let
= 2. Then,
4=6
=6
2
\ / x++ 4 = 118
8
2Jx
(if
- 2(i) - 15 = 0
(g>2_2<g>_15=0
V
+ 44 =z 99
Jxx +
xX + 44 ==881
1
xX ==777
7
A2‐2A‐15=0
A2 - 2A - 15 = 0
( A -‐ 55)(A
) ( A++ 3)
3 )== 0
(A
0
A=
= 5, -‐33
xX
xX
..
..
So
= -‐ 33.. Solvmg
Solvmg these
So 8 := 5 or E =
these equations
equations
14. (E]
14.‑
6
2
20‐fi=§fi+10
20 - ..rx
= Fx+10
3
18. [zJThere
w o ways
There are
are ttwo
ways to approach
approach this
this
problem. The
problem.
The faster
faster way
way is to factor
factor the
the
5
102§fi
10=
3../x
3
numerator
numerator first.
first.
e
fi
6 =zv'x
3
= Xx
366=
2
xx2_
- 4x+3z4
4x + 3 =
4
x -‐ 1l
(x‐3)jJ//1‘)':4
3 )__(,x.----lj
=4
Square both
both sides
sides of the
15. 11]
Square
the equation.
equation.
(X -
M
.x--T
2
x -‐ 3 =
: 4
+ y)2 =
(Jx2 +
(x +y)2
= («33
+y2
y2 +16)
+ 16)2
x=
=7
x2+2xy+y2
x2 + 2xy + y2=x2+y2+16
= x 2 + y2+ 16
2xy
2xy = 16
xy =
= 88
xy
16.
6
gives x = 30
gives
8 . Since the
30 and
and x = -‐ 118.
the question
question
specifies that
specifies
that x
x > 0, the answer
answer is 30.
The second
second way
The
r i d of the
the fraction
way is to get
get rid
fraction
by multiplying
multiplying both
first by
both sides
1 and
and
sides by x
x -‐ 1
then
then factor
factor later.
later.
2 - 4x + 3 _
xx2‐4x+3
=4
x -‐ 1
4
x2‐4x+3:4(x‐1)
x2 - 4x + 3 = 4 (x - 1)
[g Cross
both sides.
sides.
Cross multiply
multiply and
and expand
expand both
4(2x‐
1 )z
) ( x-‐ 22))
4(2x
- 1)
= ((xx ++ 22)(x
8
: xx22 -‐ 44
Bxx- ‐ 44=
0=x2‐8x
0
= x 2 - Bx
0
0=
= xx((xx -‐ 88))
x2‐4x+3=4x‐4
x 2 - 4x + 3 = 4x - 4
x2-8x+7=0
x 2 - Bx + 7 = 0
(( Xx -‐ 77))((Xx- ‐ 1
=0
1)) =
xX ==00,8
,8
xX ==77,1
,1
Neither
are false solutions,
Neither are
set
solutions, so
so the
the solution
solution set
is
is {0,8}.
Now,
Now, x =
solution since
since it causes
= 1 is a false solution
causes
division by 0. Therefore,
division
= 7.
Therefore, the
the solution
solution is x =
290
THE COLLEGE
PANDA
COLLEGE PANDA
calculator question,
no calculator
a no
this is a
Because this
19. [}] Because
question,
can
strategy. You can
valid strategy.
a valid
check is a
and check
guess and
guess
following:
the following:
do the
also do
also
23.
II]
m2g ‘- µm1g
= "128
a=
l e g
m1+
"12
+ m2
m1
x20:4
9) = 8x
8x44
x2 (x4 ‐- 9)
a(m1+
ymlg
m2g ‐- µm1g
= ng
a(m1 + m2) =
= ‐- yµm1g
m2g =
a(m1 + m2) ‐- "128
a("11+
m1g
x - 9x - 8x =
x6‐9x2‐8x4
:00
2
- 8x -‐ 9) =
x 2 (x 4 ‐8x2
x2(x4
z 00
6
2
4
ng
a(m1 + m2)
m2g ‐- a(m1+
m1g
m
lg
x2(x2‐9)(x2+1)=0
x2 (x 2 - 9)(x 2 + 1) = 0
x2(x‐+‐3)(x
0
1) = 0
3)(x 2 + l)
x 2 (x + 3)(x ‐- 3)(x2
24.
the
out, the
cancel out,
2's cancel
the 2’s
Because the
[JJBecause
same.
the same.
stays the
acceleration stays
acceleration
equation
the equation
be 3 for the
must be
0, x must
Because
Because x > 0,
true.
be true.
above to be
above
_ 2 n g ‐ y(2m1)g = 207123 ‐ umig)
2m1+2m2
20711 + 7712)
y values.
x and J/
given
20. II]
20‘
C Plug
Plug iinn tthe
h e 81
‘venxand
values.
anew ‑
:=” oaold
ld
kx 2 + 5
y + 2kx = kx2
23+
= k(3)
k(3)22 + 5
2(k) (3) =
23 + 2(k)(3)
2
+ 6k
6 k== 99kk+ 5
233+
25.
‐- 33kk = ‐- 118
8
k =: 6
21.
E_x+12
X + 12
X
26.
42
Then
side. Then
hand side.
left hand
the left
expand the
First, expand
First,
[}QJ
combine
factor .
and factor.
terms and
like terms
combine like
4
2 x=: 66xx ++7722
42x
3
6 x = 7722
36x
x=
: 2
x6
[]J
3(x
3
( x-‐ 22y)
y ) -‐ 33z2 =
: 00
3
y --3 2
3z == 0
3xx -‐ 66y
3x
6y+32
+ 3z
3x =: 6y
xX =
= 2y
2y+z
+z
multiply .
Cross multiply.
II] Cross
6
= fl
((xx+
+ ll ))((xx-‐ 22)) = 77xx -‐ 118
8
2
- x - 2 = 7x - 18
xxz‐x‐2=7x‐18
6
x2‐8x+16:0
x 2 - Bx+ 16 = 0
2
(x‐4)2=0
(x - 4)2 = 0
N o w , =‐ ‐ ‐ == 3.
3
Now,
22.‑
22. II]
Therefore,
and
Therefore, x = 4 and
7x
‐
18=
7(4)
‐
18=
18 = 10.
7x - 18 =
c+1
d = a( c~ l)
d=“<
24)
a
C+
l)
=a(c~
i§=“(z_4>
1
cC + 11
5
2 : 247
1 2=: cC +
+ 11
12
l11l =
z cC
291
291
CHAPTER
CHAPTER 30 ANSWERS
ANSWERS TO THE EXERCISES
EXERCISES
IRJ
27. El We can
can either
answer choices
either plug
plug in the
the answer
choices
or solve
solve algebraically.
algebraically. Plugging
Plugging in the
the answer
answer
choices
choices is more
more efficient
efficient here,
here, but
but since
since that’s
that's
self‐explanatory,
solve algebraically.
algebraically.
self-explanatory, let’s
let's solve
First,
both sides.
First, square
square both
sides.
is false.
A.. For
choice B,
B,
false. We can
can eliminate
eliminate A
For choice
2,/2 =
Js, so
the value
2\/2
= \/§,
so ifif that's
that's the
value of Jx
\/x -‐ 10,
then
have to equal
we plug
then x would
would have
equal 18. If we
plug
18 into
into the
the equation,
equation, the
the left hand
hand side
x = 18
side is
\/18
10=
the right
right hand
side is
is
J 18 -‐ 10
= 2\/2
2 ,/2 and
and the
hand side
m‐
/18 - fz=3\f2‐
,/2 = 3 ,/2 - «
,/25=: 2NE.
,/2 . Sinceboth
Since both
sides
be the
the answer.
sides match,
match, choice
choice B must
must be
answer .
(2/x)2
= (x - 3)2
(2\/§)2:(x‐3)2
4x=x2‐6x+9
4x
= x 2 - 6x + 9
30.
IT]
O=x2‐10x+9
0 = x 2 - l0 x + 9
= ((xx ‐- 1l ))(x( x -‐ 99))
0=
xy2+x‐y2‐1=0
xy 2 + X - y2 - 1 = 0
(y2+ 1) - (y2 + 1) = 0
X(y2+1)‐(y2+1)=0
(y2+ 1) (X - 1) = 0
(y2+1)(x‐1):0
X
=
x = 11,9
,9
X
Since 1 is a
a false
solution, the
the only
Since
false solution,
only value
value of x
that satisfies
satisfies the
the equation
that
equation is 9.
Since y2
Since
+ 11is
always positive,
equal
y2+
is always
positive , x must
must equal
1.
28.
28.[Il
31.
4
4
_
=
xx 22- ‐ 6
6xx++99 _
44
(x
- 3) 2
(Y‐3)2
9
[I]
Divide
both sides
Divide both
sides by P and
and take
take the
the tth
tth
root of both
root
both sides.
sides.
V =
P((11- ‐ r)
: P
r)’1
=
=99
~p = (1 -‐r)1
r) 1
g=(x‐3)2
; = (x - 3)
2
fig:
±
'9= / (x-
~=
1- r
(x‐3)2
3)
2
,IV
r = l - Vp
i§=x‐3
3
2
±- = x - 3
32.
29.
I]] We can
can use
use the
the answer
answer choices
choices to
[I]
From the
the previous
From
question, we
we know
know
previous question,
_
tthat
h a rt r=‐ 11‐-
backsolve or we can
backsolve
can solve
solve the
the equation
equation
algebraically. We'll first solve
solve it algebraically
algebraically.
algebraically
by squaring
squaring both
both sides.
sides.
,IV
i/V
zP =‐_12..
$.BecauseVishalfP,
V
p·
Because Vis- half P, p
2
Thus,r:1‐{‘/gz0.l3
= 1 - if"{:=::::
0.13
Thus , r
(✓x
- 10 ) 2 = (vx - 12)2
(x/x‐10>2=(f‐\/§)2
(Jx)2
/x)(n /h5 ) + ( /2
fi f -‐ 2(
aw
x / )2
if
x‐
10= :x -x‐zx/fln
x - 10
2../h + 2
xx -‐ 10
1 0=
:
-‐12=‐2\/fl
12 = - 2./ix
6 = ...fix,
6=\/2_x
3 6=: 22xx
36
1 8== Xx
18
Let's
we wanted
wanted to test
Let’s say we
answer
test the
the answer
choices
instead . H
How
choices instead.
o w would
that? For
would we do that?
For
choice
../6is the
the value
choice A, if
if \/5
\/xx -‐ 10,
10, then
then x
value of J
would have
have to equal
equal 16. If we
we plug
z 16
16
would
plug x =
into
the equation,
equation, we
we get
into the
get 44 =
= 4 -‐ \/2,
,/2, which
which
292
292
V
1
THE
THE COLLEGE
COLLEGE PANDA
PANDA
Chapter 9: More Equation
Strategies
Equation Solving
Solving Strategies
CHAPTER EXERCISE:
EXERCISE:
1. I55 I30
30 ((x3
+
1.
x3 +
+ ~§x>
~éxz
x2 +
x)
haven’t
assumed that
haven't assumed
that a and
and b are
are positive
positive
here.
here . But of the
the answer
answer choices,
choices, only
only 6 is aa
possible
= 3 and
and bb =
= 22 or
possible value
value of ab
ab (when
(when a =
when
z -‐33 and
= -‐ 22).
).
when a =
and b =
= 30x
30x33 +
+ 5x
5x22 +
+ 20x.
=
20x.
Therefore,aa = 30,
30,bb = 5, and
z 20.
Therefore,
and c =
+ bb ++cc =
: 555.
5.
a+
2.. [Qj
E] For
For an
to have
an equation
equation to
have infinitely
infinitely many
many
solutions,
sides must
must be
solutions, both
both sides
be equivalent.
equivalent.
5.
0
Expand both sides of the equation.
3
2
1
-- x - 4 = 4 -2
Comparing
sides, ~a
ga = ~g
Comparing the
the terms
terms on
on both
both sides,
and 3 =
= 9b. Solving
Solving these
these equations,
equations, we
we get
get
and
1
a
4
a=
= 4 and
Therefore, b
5 = I= 12.
and b = 5.. Therefore,
3
‐ Ex‐ 4‐ ‐ 2x
3
Let’s
rearrange the
right side
that the
the terms
terms
Let's rearrange
the right
side so
so that
line
left side.
side.
line up with
with the
the left
3.. [Qj
[E For
For an
no solutions,
solutions, the
an equation
equation to have
have no
the
coefficients
coefficients of the
u s t be
be the
same
the x terms
terms m
must
the same
on either
side but
m u s t be
be
either side
but the
the constants
constants must
different. If we
different.
we expand
expand the
the right
right side,
side, we
we get
get
1
1
-‐ ‐xx -‐ 4 =-: ‐ ‐xx+4 4
2
2 +
Now it’s
Now
easy to see
see that
that the
the
it's easy
the coefficients
coefficients of the
x terms
terms are
same but
the constants
constants are
are
are the
the same
but the
different
the equation
different ( -‐ 4 vs. 4). Therefore,
Therefore, the
equation
a
axx-‐ b =
= 66xx + 3
Therefore, aa =
= 6 and
and b f=
742 ‐ 33.. Only
Only answer
answer
Therefore,
choice D satisfies
satisfies these
these conditions.
conditions. Choice
choice
Choice C
would
result in infinitely
many solutions
solutions
would result
infinitely many
since both
both sides
would be
beequivalent.
since
sides would
equivalent.
has
solutions.
has no
no solutions.
6.
0
= (p
(F7 +
+ q)(P
Q)- We
4.. Remember,P2
Remember, p2 -‐ £72
q2 =
q)(p -‐ q).
We
can
take
can apply
apply this
this factorization
factorization here
here once
once we
we take
out aa 2:
18x22 -‐ 8 = 2(9x2
= 2(3x ++ 2)(3x
2)(3x -‐ 2),
2),
18x
2(9x 2 -‐ 4) =
which
equals 2(ax
2(ax + b)(ax
b)(ax -‐ b).
b). Comparing
Comparing
which equals
the coefficients,
coefficients, a = 3 and
and b =
: 2. Therefore,
Therefore,
the
ab =
: (3)
(3)(2)
:
6.
Note
that
this
factorization
(2) = Note that this factorization
method assumes
assumes that
the constants
are
method
that the
constants are
positive,
but that's
that’s okay
okay since
all the
the answer
answer
positive, but
since all
choices
It’s possible
that aa =
z -‐33
choices are
are positive.
positive. It's
possible that
and b =
: 2 or a =
= 3 and
and
and b = -‐ 22,, for instance,
instance,
but
but these
these are
are cases
cases that
that you
you generally
generally don't
don't
need
need to worry
worry about
about for this
this factorization.
factorization .
[QjFor an equation
to have no solutions, the
coefficients
terms must
m u s t be
be the
same
coefficients of the
the x terms
the same
on either
side but
but the
constants must
must be
either side
the constants
different. First,
First, let's
let’s expand
left side.
side.
different.
expand the
the left
3 x+
+3
a x=
= 1122-‐ 77xx
3x
3aa-‐ 22ax
3 a-‐ 22ax
a x = 112
2 -‐ 1lOx
0x
3a
Comparing
Comparing the
the coefficients
coefficients of the
the x terms,
terms,
a=
0 , a = 5. Note
Note that
is N
O T equal
-‐ 22a
= -‐ 110,
that a is
NOT
equal
to 4 since
o t for the
since the
the goal
goal is n
not
the constants
constants to
be the
same..
the same
7.
[II Expand
the right side.
(2x + 3)(ax
=12x2
15
3)(ax -‐ 5) =
12x2 + bx -‐ 15
If you
this factorization
factorization and
and
you didn’t
didn't use
use this
instead expanded
expanded the
the right
right side,
side, you
you
instead
would’ve gotten
gotten
would've
18x2 ‐ 8 = 2a2x2 ‐
1
2
1
x
2
xX--‐ 5X
x -‐ 44 =
‐ 44 -‐' 5xx
2ax2
10x -‐ 15 = 12x2
2ax 2 + 3ax -‐ lOx
12x2 + bx
bx -‐ 15
2ax2
0 ) x -‐ 15
1 5=
b x‐- 15
15
2ax 2 + (311
(3a -‐ 1lO)x
= 1212
12x2 + bx
Comparing
12 and
Comparing both
both sides,
sides, 2a
2a = 12
and
b=
which yields
and
= 3a
3a ‐- 10, which
yields a =z 6 and
b = 33aa -‐ 1100 =: 33(6)
( 6 )-‐ 1100=: 88..
2172
Comparing the
either side,
side,
Comparing
the coefficients
coefficients on either
18 = 2a2
18
2a2 and
and 8 = 2b2.
2b2 . Solving
Solving these
these equations
equations
gives
$33 and
= ±
:l:2.
you can
can see,
see, we
we
gives a
a := ±
and b =
2. As
As you
293
CHAPTER 30 ANSWERS
CHAPTER
ANSWERS TO THE
THE EXERCISES
EXERCISES
8. j 49 IExpand
Expand the
side:
the left side:
13. E]
@JMultiply
Multiply both
both sides
sides by
by xyp.
xyp.
x2+6xy+9y2=x2+9y2+42
x2 + 6xy + 9y 2 = x2 + 9y 2 + 42
_1 + 1_ : _1
-+-=xx
/xl+6xy+9f=xz+9
+ 6xy + JI = / + JI +42
+ 42
w+w=w
x2y2
x2y2=
= 49
yp =
: X
__1E_
p
yy -‐ p
9. [fil For an
9.
an equation
equation to
to have
have infinitely
infinitely many
many
solutions, both
both sides
sides must
solutions,
must beequivalent.
be equivalent.
First, let’s
let's expand
First,
expand the
right side
the right
side of the
the
equation:
equation:
14.
LE]
@]Expand
Expand the
the left side
side of the
the equation.
equation.
(x3 + kx2 - 3) (x - 2)
(x3+kx2‐3)(x‐2)
6xx =: xx-‐ 66nx
6
nx+
+33xx
6x
6x = 4x -‐ 6nx
6nx
= x 4 + kx3 - 3x - 2x3 - 2kx2 + 6
=x4+kx3‐3x‐2x3‐2kx2+6
= x 4 + (k - 2)x 3 - 2kx 2 - 3x + 6
=x4+(k‐2)x3‐2kx2‐3x+6
2x == -‐6nx
2x
6nx
Comparing this
this to x4
Comparing
18x22 ‐- 3x
x 4 + 7x3
7x3 -‐ 18x
3x +
+ 6,
we can see that k - 2 = 77and
and ‐- 22kk =
wecanseethatk‐
= -‐ 118.
8 . lInn
both cases,
cases, k =
both
: 9.
Now
when we compare
compare the
N o w when
the coefficients
coefficients on
both
sides, we get 2 =
both sides,
= -‐ 66n,
n , which
which gives
gives
1
2
n:_‐6=
n =- 6 =-3·3.
[j
15. ~ Multiply
Multiply both
both sides
sides by
by (x +
+ 3)(x
3)(x ‐- 2). We
We
get
Multiply both
both sides
sides by
Multiply
by b.
b.
5(x
5
( x-- 22)) ‐- 22(x
( x + 33)) = aaxx -‐ b
5
2 x- ‐ 66 =
z aaxx -‐ b
5xx-‐ 1
100- ‐ 2x
3x -‐ 16
ax -‐ b
3x
16 = ax
a
b
abb++aa = aa++5Sb
a
b
=
5
b
ab Sb
a=5
S
11.
[IjMultiply
Multiply both
sides by
by x(x
x(x -‐
both sides
Comparing the
the coefficients
coefficients on
Comparing
on either
either side,
side,
a := 3andb
= 16. Therefore,
Therefore,
3 and b =
a
+ b ==33++116
6 ==119.
9.
a+b
4).
4).
((xx-‐ 4
x ( xx -‐ 44))
4)) -‐ xx ==x(
lNotice
( x -‐
INotice that
that x22‐- 11==( x(x++1 l))(x
1
16. j ;
-‐44 z=x2
x ‐- 4x
4x
2
o
(x -‐ 2)
2)22
0=
= (x
4
( x++ l1)+2(x) + 2 ( x ‐ 1)
1 )=: 335
5
4(x
4
5
4xx++ 44 ++ 2
2xx-‐ 22 ==335
6
5
6xx++ 22 z=335
6
6xx = 3333
We can
can see that
that x = 2.
0
1
1))o
onn
right hand
hand side.
side. It’s
the right
It's then
then easy
easy to see
see that
that
we should
should multiply
we
sides by
multiply both
both sides
by
((xx + l ))(x
( x -‐ l1).
).
0
z x2
0=
x 2 -‐ 4x
4x +
+ 44
12.
p
p
YP + xp = xy
- xxpp
yypp = xxyy ~
yp
=
x(y
y p = X ( l / -~ pp))
6xy
= 42
6xy =
xy
xy = 77
10.
yy
Expanding
side,
Expanding the right
right side,
4x2+mx+9
z4x2+4nx+n2
4x2 + mx + 9 =
4x2 + 4nx + n2
xV2
=
_ 22
11
Comparing
Comparing both
both sides,
we see
see that
sides, we
that
= 4n
= 1n122 and
and m
m =
4n
9
9:
Therefore, n = -‐33 and
and m =
Therefore,
= 4(‐3)
2
4( -3) = -‐ 112
m
+n =
: ‐ -1 12
2 ++((‐- 33)) = ‐- 115
5
m+
294
THE COLLEGE
PANDA
THE
COLLEGE PANDA
17. [[) Expand
hand side.
Expand the
the left
left hand
side.
(2x -‐ b)(
7x +
14x 2 -‐ ex
b)(7x
+ b) =
=14x2
cx -‐ 16
16
2
-‐ ex
14x
b2 =
14x22 +
+ 2bx
Zinc -‐ 7bx - b2
= 14x
143:2
cx -‐ 16
b2 = 14x
14x
14x22 -‐ Sbx
5bx -‐ b2
14x22 -‐ ex
cx -‐ 16
Comparing both
can see
Comparing
both sides,
sides, we
we can
see that
that b =
=4
(b cannot
0).
cannot be
be -‐44 because
because b > O).
2 Sb
5b = 5(4) =
= 20.
ec =
18.
@]
Multiply both
by (n -‐ 1l))(( n +
lMultiply
both sides
sides by(n
+ 1).
1).
3(n
2n (n -‐ 1)
(11+
l )( n -‐ 1)
3(n +
+ 1)
1) +
+2n(n
1) = 33(n
+1)(n
1)
3n
+ 3 + 2n 2 - 2n =
3(112 - 1)
3n+3+2n2‐2n
=3(n2‐1)
2
2n2+n+3=3n2‐3
2n
+ n + 3 = 3n 2 - 3
2
00 =
- ‐11
= nn2
n- ‐66
00 := ((n
+22))
n ‐- 33)(
)(n+
n = 33 oorr -‐ 22.. BBecause
e c a u s ne n>>0,
0 ,nn== 33..
295
295
CHAPTER 30 ANSWERS
CHAPTER
ANSWERS TO
TO THE
THE EXERCISES
EXERCISES
Chapter 10: Systemsof
Chapter
Systems of Equations
Equations
CHAPTER EXERCISE:
CHAPTER
1. [fil Substituting
1.
Substituting the
the
the second
second equation
equation into
into the
;
6.
6.
first,
first,
w o lines
[I] IfIf the
the ttwo
lines intersect
intersect at
at the
the point
point (2,
(2, 8),
8),
then
solution to
system.
then (2,8)
(2,8 ) is a
a solution
to the
the system.
Plugging the
Plugging
the point
point into
into the
the equation
equation of
of the
the
second line,
second
line, we can
can solve
solve for
for b,
b,
3 (1 - 3y) - 5y =- ‐ 111
3(1~3y)‐5y=
1
3 -‐ 99yy -‐ 5Syy =
z ‐- 111
1
3 -‐ 114y
3
4 y== ‐- 1111
yy =
z ‘- bbx"
8 == ‐b(2)
- b(2)
8
‐- 4 =
=b
‐- 114y
4 y == ‐- 1144
l
yy =
=1
Plugging the
Plugging
the point
point into
into the
the equation
equation of
of the
the
Finally
Finally,, x =
z 11 -‐ 3(1) =
= -‐ 22..
line,
first line,
[QJFrom
2. E]
From the
the first equation,
= 20
equation, 31
y=
20 -‐ 2x.
2x.
Plugging this
Plugging
this into
into the
the second
second equation,
equation,
ax +
+b
b
yy =
: ax
8 = a(2)
a (2) -‐ 4
6xx -‐ 55(20
6
( 2 0-‐ 22x)
x ) = 112
2
1 2=: 22a“
12
6
6 =: aa
6xx -‐ 1100
lOx
6
0 0+
+1
0 x=: 112
2
16x
: 112
16x =
x =
z 7
7.
The two graphs do not intersect at all, so
there are
there
are no
no solutions.
solutions.
We
already know
Wealready
know the
at this
this
the answer
answer is
is (D)
(D) at
point , but
point,
z 20
‐
2(7)
=
6.
but just
just in case,
case, y =
20 - 2(7) = 6.
3.
0
8.
[1]From
From the
the first
first equation,
equation , we
we can
can isolate
isolate yy to
to
get y = -‐ 5Sxx ‐- 2. Substituting
get
Substituting this
this into
into the
the
second
equation,
second equation ,
[1]Add
Add the
w o equations
the ttwo
equations to
to get
get
7x -‐ 7y
7x
7y = 35. Dividing
both sides
sides by
by 7,
Dividing both
7,
5. We can
x ‐- y == 5.We
can multiply
multiply both
both sides
sides by
by ~1
- 1
to
get
y
x
=
5.
togety‐x=‐5.
2(2x
= 33 -‐ 3(‐5x
2(2x -‐ 1)
1) =
3(- Sx -‐ 2)
2)
4
3 3+ +
15
x ++66
4xx-‐ 22 :=
15x
4. ~ The
The fastest
fastest way
way to do
do this
to
this problem
problem is
is to
subtract the
the second
subtract
second equation
equation from
from the
the first,
first,
which
yields x +
+y =
z 9.
9.
which yields
4x ‐- 2 := 15x +
+9
11x
-‐ 1ll1 =
= ll
x
_1
z
x
- 1=X
[I] In the
the first equation,
we can
m o v e 3x
3x to
to
equation , we
can move
Finally,
_5( _1) _ 2 = 3.
Finally,yy z
= - 5( - 1) - 2 = 3.
5.
the right
right hand
the
: ‐- 5Sxx +
+ 8.
8.
hand side
side to
to get
get y =
Substituting this
Substituting
into the
the second
second equation,
equation,
this into
2((-‐ 55xx +
+ 8)
8) =
”- 33xx +
+2
=
z
-‐ 33xx ‐- 10x
lOx +
+ 16
16 =
:
-~13x
13x =
x Z
x=
9_
9. [1]Divide
Divide the
the first
first equation
equation by
by 2
2 to
to get
get
x ‐- 2y := 4. We can’t
can't get
get the
the coefficients
coefficients to
to
match
match ((-‐ 2 vs.
vs . 2
2 for
for the
the y’s).
y's). Therefore,
Therefore, the
the
system
solution.. In
system has
has one
one solution
In fact,
fact, we
we can
can even
even
solve
w o equations
solve this
this system
system by
by adding
adding the
the ttwo
equations
to get
y=
= O.
get 2x = 8,
8, x = 4,
4, which
which makes
makes y
0.
-‘ 110
0
‐- 110
0
6
-‐ 226
2
2
10.
Then, y = ‐5(2)
Then,y
- 5 (2) + 8 z
= ‐- 22.. Finally,
Finally,
xy
=
(2)(
2)
=
4.
xy = <2>(‐2) ‐ 4 .
the
[I] To get
get the
the same
same coefficients,
coefficients, multiply
multiply the
first equation
10y =
equation by ‐2
- 2 to get
get ‐- 44xx +
+ lOy
= -‐2a.
2a.
Now
N
o w we
a = ‐- 88,, a
we can
can see
see that
that ‐- 22a
a=
= 4.
4.
296
THE
COLLEGE PANDA
THE COLLEGE
PANDA
11.
0
the
17. E]
~ Plugging
Plugging the
the first
first equation
equation into
into the
second,
second,
Multiply
equation by
Multiply the
the first
first equation
by -3
‐3 to
to get
get
-‐3ax
3ax -‐ 6y =
15. The
= -‐15.
The constant
constant a cannot
cannot be
be
-‐ 1.
equation 's
1. Otherwise,
Otherwise, the
the second
second equation’s
coefficients
equal to the
coefficients would
would then
then be
be equal
the first
equation's coefficients,
equation’s
coefficients, resulting
resulting in aa system
system
v'4x
m
‐ -( fi(v1x
++
3 3)) == 33
z fi -‐ fiVX
‐ -33== 33
2/x
/x
fi ==6
6
x z=336
6
X
with
with no
no solution
solution..
12. @ First,
First, multiply
multiply the
the fust
first equation
equation by
by 33 to
to
get
rid
of
the
fraction:
12x
y
=
24.
Next,
get rid the fraction:
‐ = ‐24. Next,
substitute
substitute the
the second
second equation
equation into
into the
the fust,
first,
Therefore,
\/3_6 +
+3=
: 9.
Therefore, y = v'36
12x -‐ (4x
(4x + 16)
16) =
= -‐ 224
4
18.
12x
1 2 x- ‐4x
4 x- ‐ 16
1 6=: -‐ 224
4
medium, and
and large
large jars,
respectively. Based
Based
medium,
jars, respectively.
on the
information, we
can create
on
the information,
we can
create the
the
following
following ttwo
w o equations
equations::
Bx
8 x = -‐ 8
x
X
== -‐ 1
Finally,y
4(‐1)
+ 116
6=
= 12.
y = 4(1) +
Finally,
13.
16s=2m+1
= 2m + I
16s
4
s +m
m = 1I
4s+
[IQ]We
Wecan
can isolate
isolate x in the
the second
second equation
equation to
to
Substituting this
get
get x =
= y -‐ 18. Substituting
this into
into the
the first
first
equation,
equation,
the weight
weight of the
the large
terms of
To get
get the
large jar
jar in terms
the
weight
of
the
small
jar,
we
need
get rid
the weight the small
we need to
to get
rid
the weight
weight of the
the medium
of m,
m, the
medium jar
jar.. We
Wecould
could
certainly use
elimination, but
here, we'll
certainly
use elimination,
but here,
we’ll use
use
substitution. Isolating
the second
substitution.
Isolating m
m in the
second
into
equation, m
equation,
m = I -‐ 4s. Substituting
Substituting this
this into
the first
first equation,
equation, we
we get
the
get
y3]=
= O.S
(y - 18) +
0.5(y‐18)
+ 114
4
. 5 y-‐ 99 ++114
4
yy ==00.Sy
O.5y=5
0.Sy
=5
0
y ==110
14.
16s=2(1~4s)+l
16s = 2(/ - 4s) + /
Cf]To match
coefficients, multiply
match the
the coefficients,
multiply the
the
16S=21‐SS+1
16s
= 21- 8s + I
equation by
by 18
18to
get 6x
6x -‐ 3y =
: 72. We
We
first equation
to get
can then
can
then see
see that
that aa =
= 33 if
if the
the system
system is to have
have
no solution
solution..
no
24s=3l
24s = 31
8
8s5=: 1I
E’ Divide
Divide the
the fust
first equation
equation by
by 3 to
to get
get
15. [Q]
2y = 5. Divide
Divide the
the second
second equation
equation by
by -‐22
x -‐ 2y
to get
get x -‐ 2y
2y =
: 5. They're
They're the
the same,
same, so
so there
there
are
are an
an infinite
infinite number
number of solutions
solutions..
16.
Eight
small jars
jars are
needed to match
Eight smaJl
are needed
match the
the
weight
large jar.
weight of one
one large
E] Since
were 30
30 questions,
questions, James
James must
Since there
there were
must
19. [Q]
have
z 30. The
The points
points
have had
had 30 answers,
answers, x + y =
he
5x. The
The
he earned
earned from
from correct
correct answers
answers total
total 5x.
points
he lost
lost from
answers total
points he
from incorrect
incorrect answers
total 2y.
Therefore, 5x
5x -‐ 2y
231 = 59.
Therefore,
For aa system
system to
to have
have infinitely
infinitely many
many
Cf]For
solutions,
solutions, the
the equations
equations must
must essentially
essentially be
be
the same.
same. Looking
Looking at the
the constants,
constants, we can
can
the
make them
them match
match by
by multiplying
multiplying the
the second
second
make
equation by 2. The
The equations
equations then
then look
look like
like
equation
this::
this
Leta and
points you
you get
get
be the
the number
number points
20. @] Leta
and b be
for hitting
hitting regions
A and
and B,respectively.
regions A
B, respectively.
From
information, we
form the
the
From the
the information,
we can
can form
following
t w o equations:
following two
equations :
mx -‐ 6y
6y =: 10
mx
10
4x -‐ 2ny = 10
10
4x
+ 22bb = 118
8
a+
2
a
+
b
=
2
2a + b 211
N o w it's
it’s easy
easy to
to see
see that
that m
m=
z 4 and
and 2n
Zn =
: 6,
6,
Now
.
m_ 4
n =‐ 3.
3. Finally,
Finally, ;:
n
[II Lets,
Lets, m,
and I be
be the
the weights
ll,
m, and
weights of sma
small,
‐ ;.5.
=
To solve
solve for b,
b,multiply
first equation
equation by
by 22
multiply the
the first
and
subtract to get
get 3b
= 15,
15, b = 5.
5.
and subtract
3b =
297
CHAPTER 30 ANSWERS TO THE EXERCISES
21.
[Q]Let r and
c be the number of rectangular
tables
tables and
and circular
circular tables,
tables, respectively,
respectively, at
at the
the
restaurant.
Based on
on the
restaurant. Based
the information,
information, we
we can
can
make
make the
the following
following two
t w o equations:
equations:
24.
II] To find the point(s)
where two graphs
intersect,
solve the
the system
system consisting
their
intersect, solve
consisting of their
equations . In this
this problem,
problem, that
that system
system is
equations.
y=x2‐7x+7
y = x 2 - 7x + 7
: 22xx -‐ 1
y=
4r+
+ 88cc ==1144
44
4r
rr
+
+cC ==3300
Substituting
the first equation
equation into
the
Substituting the
into the
second,
we
get
second,
get
To solve
solve for r, multiply
multiply the
the second
second equation
equation
by 8 and
and subtract
subtract to get
get -‐ 44rr =
: -‐ 996,
6 , r = 24.
2 - 7x + 7 = 2x - 1
xx2‐7x+7=2x‐1
22. ~ The
The solution
solution to the
the system
system is
is the
the
intersection
point
of
the
two
lines.
intersection point the t w o lines. Each
Each
2
- 9x + 8 = 0
xx2~9x+8:0
(x( x ‐ 1l )(x) ( x ‐ 8)=
8 ) = 00
horizontal
horizontal step
step along
along the
the grid
grid represents
represents ~2 of
xl e= o1 or
r 8
aa unit,
and each
unit, and
each vertical
vertical step
step along
along the
the grid
grid
represents
represents 1 unit.
unit. So,
So, the
the intersection
intersection point
point is
3
at( - ~, - 3).
So the
the x-coordinates
x-coordinates of the
the points
So
points of
intersection are
and 8. Since
intersection
are 1 and
Since the
the question
question
must be
already gave
(1, 1), p must
already
gave us
us the
the point
point (1,1),
be
“(74>
23.
equal to 8.
equal
[Q]From the second
equation, x = 2y.
Plugging
the first equation,
Plugging this
this into
into the
equation, we
we get
get
I
25. 9 or 16 j First,
both sides
sides of the
First, add
add 11 to both
the
second
equation to
second equation
to get
get y =: x + 11. Then
Then
substitu te this
this in for y in the
the first equation:
substitute
equation:
2 _ 2= : _1_
(2y)2
(231) _ y2
y
12
12
2 _ 2 == -1_
4y2
4y - y2
y
12
12
1_
‐ _1
3y22 =
3 y ' i-12z
x 2 - 2x = x + 11 - 1
x2‐2x:x+11‐1
2 - 3x - 10 = 0
xx2‐3x-1ozo
((xx ++ 2
) ( x- ‐ 5)
5 )=
=0
2)(x
=-‐ 22or5
x=
0r5
1
y2=l
= 36
2
y
When
+1111 = 99.. W
When
W
h e nxx== -‐ 22,,yy == -‐ 22 +
h e nxx == 55,,
The so
lutions to the
the system
y := 5 + 11 = 16. The
solutions
system
are then
then (‐2,9)
(- 2, 9) and
and (5,16).
(5, 16). Therefore,
the
are
Therefore, the
possible values
possible
values of y are
are 9
9 and
and 16.
1
yy =
‐ i±-g
6
1
Therefore,
Therefore, the
the values
values of y1 and
and y2
y; are
are -‐2 and
and
6
1
6.
6
298
THE COLLEGE
COLLEGE PANDA
THE
PANDA
Chapter11:
Chapter
11: Inequalities
Inequalities
CHAPTE R EXERCISE :
1.‑
-‐ xx ‐- 44>>44x
x ‐- 114
4
‐- 55xx > ‐-110
0
x<
<2
the answer
Of the
answer choices,
choices, only
only -‐11 is
is aa solution.
solution.
2.
El Multiply
to get
r i d of
fractions..
2. [QJ
Multiply both
both sides
sides by
by 44 to
get rid
of the
the fractions
1
3
-Zx‐4>§x‐10
x - 4 > - x - 10
4
2
3xx-‐ 1166 > 22xx -‐ 4400
3
xX >> ‐-224
4
3.
3.
The shaded
shaded region
below the
the horizontal
: 3,
above
[g The
region falls below
horizontal line
line 31
y=
3, soy
soy < 3.
3. The
The shaded
shaded region
region also
also stays
stays above
y=
soy
= x,
X, SO
y > x.
X.
4.
Let’s say
say Jerry's
Jerry's estimate,
estimate, m,
m, is 100 marbles.
marbles.
[fil Let's
5.
5.
[I] Setting
Setting up
up the
the inequality,
inequality,
If the
the actual
actual number
of marb
marbles
is within
within 10
If
number of
les is
10 of
of that
that
estimate,
estimate, then
then the
the actual
number must
m u s t be
be at
at least
least 90
m ‐- 10
5 n
S
actual number
90 and
and at
at most
most 110. Using
Using variables,
variables, m
10 ::;
11 ::;
+ 10.
m+
lO.
M
2N
N
M ?.
12P +
+100
12P
100 2
?. ‐- 33PP -++ 970
970
15P 2
15P
?. 870
P ?.
2 58
58
6.
[zJ
3 ( n-‐ 22)) > -‐ 44(11
( n -‐ 99))
3(11
3 n-‐ 66 >>‐ -44n
n ++336
6
3n
7
2
7nn > 442
n>6
Since n11is
Since
is an
integer, the
least possible
value of
of 11
n is
7.
an integer,
the least
possib le value
is 7.
7.
The shaded
shaded region
region is
is below
below the
the horizontal
horizontal line
line y3;=
‐ 3 . Therefore,
Therefore,
[fil The
= 33 but
but above
above the
the horizontal
horizontal line
line y =
= -3.
y2
?. ‐3
- 3and
and y 5
::;3.
3.
8.
8.
16 hours. Since
spends on
the bus
bus is
2 hours
he spends
spends on
the train
train is
is %
[I] The
The time
time Harry
Harry spends
on the
is ~
hours and
and the
the time
time he
on the
hours. Since
y
X
16
8
16
the
the total
total number
number of hours
hours is never
never greater
greater than
than 1, z
1.
- +
+ y- S::;1.
X
299
299
y
CHAPTER 30 ANSWERS TO THE EXERCISES
9.
10.
[I]
U the
the distributor
distributor contracts
If
u t to Company
u t to
for
contracts oout
Company A
A for
for xx hours,
hours, then
then it
it contracts
contracts oout
to Company
Company B
B for
10 ‐- x hours.
10
Company A then
produces
80x
cartons
and
Company
B
produces
140(10
‐
x)
cartons.
hours. Company
then produces
cartons and Company B produces 140(10 - x ) cartons.
Setting up
Setting
up the
the inequality,
inequality,
80x +
+140(10
1,100
140(10 ‐- x) > 1,100
[El
[QJPlug
Plug in x =
=
l,1, y z= 20
to get
15+
> a.
20 into
into the
the first
first inequality
inequality to
get 20
20 >
> 15
+ a,
a, 55 >
a. Do
Do the
the same
same for
for the
the second
second
inequality
15 < b.
is less
15. The
between
inequality to get
get 20
20 < 55 + b,
b, 15
b. 50,
So, a
a is
less than
than 5
5 and
and b
bisis greater
greater than
than 15.
The difference
difference between
the
wo m
u s t be
m o r e than
= 10.
10. Among
12 is
is the
only one
greater than
the ttwo
must
be more
than 15 ‐- 5 =
Among the
the answer
answer choices,
choices, 12
the only
one that
that is
is greater
than
10.
11. E,
[QJThe line
= ~x
2x +
+2
line going
going from
from the
the bottom-left
bottom-left to
to the
the top-right
top -right must
must be
beyy =
2 and
and the
the line
line going
going from
from the
the
top-left to the
the bottom-right
top-left
= -‐ 22xx +
+ 55 (based
bottom-right must
must be
beyy =
(based on
on the
the slopes
slopes and
and y‐intercepts).
y-intercepts). Answer
Answer (D)
(D)
correctly shades
shades in the
correctly
the region
: ~2xx +
+ 22 and
and below
region above
above y =
below yy = ‐- 22xx +
+ 5.
5.
12.
[I] One
One manicure
manicure takes
takes 1/3
hour. One
One pedicure
she
1 /3 of an
an hour.
pedicure takes
takes 11//22 an
an hour.
hour . The
The total
total number
number of
of hours
hours she
1
1
spends doing
doing manicures
spends
must be
be less
30, so
so ~m
3m +
+ 5pp ~
5 30.
manicures and
and pedicures
pedicures must
less than
than or
or equal
equal to
to 30,
30. She
She earns
earns
the manicures
manicures and
25m for the
pedicures. Altogether,
Altogether, 25m
and 40p for the
the pedicures.
25m +
+ 40}:
40p 2
2: 900.
900.
13. |E
[Q]From
From the given
-+- 12. Subtracting
Subtracting 12
12 5
which
given inequality,
inequality, x 5
~ 3k
3k +
12 from
from both
both sides
sides gives
gives x
x ‐- 12
~ 3k,
3k, which
confirms that
that I is always
confirms
always true.
true.
From the
the given
given inequality,
inequality, 3k + 12
From
2 k, which
means 2k 2
2, k
u s t also
12 2:
which means
2: -‐ 112,
k2
2: ‐- 66,, so
so 11
II m
must
also be
be true.
true.
From the
the given
given inequality,
inequality, k 5
From
I“ must
~ x. Subtracting
Subtracting k from
from both
both sides
sides gives
gives 0
OS
~ x
x -‐ k.
k. Therefore,
Therefore, III
must also
also
be true
be
true..
First,
14.13~ < x < ?~ ILet’s
Let's solve
solve these
these separately.
separately. First,
2
20
‐30 < -‐ 22xx ++ 4
-3
0 < -‐ 66xx + 1
-‐ 220
122
‐- 332
2 << -‐ 66xx
16
3
-L" >
>xx
Now
N o w for the
the second
second part,
part,
9
-‐2x
2x + 44 << -‐ ‐
2
-‐ 44xx + 88 < -‐ 9
7
-‐ 44xx < -‐ 117
>_
x >-
x4
17
4
17
16
9
1
17
16
9
10
Putting the
the ttwo
Putting
w o results
together, I < x < 3. . Therefore,
Therefore, Z < xx -‐ 2
results together,
2 < 30..
4
4
3
3
15. El
[QJIf the
the area
area is at
at least
least 300, then
2 300. The
The perimeter
+ 2y,
2y, so
then xy
xy 2:
perimeter of
of the
the rectangular
rectangular garden
garden is
is 2x
2x +
so
2x +
2 70, which
+ 2y
2y 2:
which reduces
reduces to x +
+ y ~2 35.
35.
16. ~ I is nnot
always true
true because
o t always
values. Take a
= 2
because of negative
negative values.
a z
= ‐5
- 5 and
and b
b =
2 for
for example.
example. a
a <
< b,
b, but
but
a > b2.
b2 . II is
is definitely
definitely true.
a
I I is
It’s the
true. It’s
It's the
the equivalent
equivalent of
of multiplying
multiplying both
both sides
sides by
by 2.
2. IIII
is also
also true.
true. It's
the
equivalent of multiplying
equivalent
which necessitates
necessitates a
multiplying both
both sides
sides by -‐ 11,, which
a sign
sign change.
change .
300
PANDA
COLLEGE PANDA
THE COLLEGE
THE
Word Problems
Chapter 12: Word
Chapter
Problems
CHAPTER EXERCISE:
CHAPTER
EXERCISE:
y
and y
of salmon
pounds of
2x pounds
bought 2x
Susie bought
[Q]Susie
6. E]
6.
salmon and
then
cost is
total cost
The total
pounds of
pounds
of trout.
trout. The
is then
the sum
1. [Q]
E] The
The square
square of the
sum of x and
and y is
and y is
product of x and
The product
(x + y)2. The
is xy.
xy. The
The
the difference
asks for the
question
question asks
difference:: (x +
+ y)
y)22 -‐ xy
77
Sy=
(3.50)(2x)
= 77
(3.50)(2x) + 5y
213
h ++ Sy
w =z 77
w
7x
X ==33(X
988-‐ X
9
( X ‐- 110)
0)
98 _ X = 3X _ 30
98 - X = 3X - 30
plug
can plug
we can
integers, we
be integers,
must be
and y must
Since x and
Since
above to
equation above
the equation
into the
choice into
answer choice
each
to
each answer
value for x. When
an integer
get an
we get
see ifif we
see
integer value
When
which is
~ 8.14, which
example, x z
4, for example,
= 4,
y=
is not
n o t an
an
When
7. When
be 7.
out to be
turns out
answer turns
integer.
The answer
integer. The
6.
y == 7,x
7,x =: 6.
4X = -‐ 128
-~4X
X
= 32
32
X =
contains
alloy contains
nickel alloy
The 35% nickel
7. ~ The
nickel
of nickel.
0.7 grams
2(0.35) =
= 0.7
grams of
nickel. The
The x%
x°/o nickel
vx + 5s = 9
J;
nickel.
6 = 0.06x grams
~ ·-6
contains £6
alloy contains
alloy
grams of nickel.
1 0
flvx =
: 4
X
x
=
z 16
16
2=
xX +
+2
Z 18
18
can
the information,
4. 1 s or 10 )Based
Based on
on the
information, we
we can
Now
4x +
equation 4x
the equation
form the
form
+ lOy =
: 60. N
o w it's
it's
Since x and
check. Since
and check.
guess and
just
just aa matter
matter of guess
and
find
we find
Jong before
be long
y are
are integers
integers,, it won't
won’t be
before we
example,
that works.
something that
something
works. For
For example,
possible solution.
one possible
= 5,
5,yy =
= 4 is one
solution.
x=
5.
5.
formed
alloys formed
these alloys
combined, these
When combined,
When
nickel alloy,
a 20% nickel
2+
+6=
= 8 grams
grams of a
alloy, which
which
nickel.
of
grams
1.6
=
8(0.20)
contain
must
must contain 8(0.20) = 1.6 grams of nickel.
get
we get
equation, we
an equation,
up an
Setting up
Setting
‘
0.7 + 0.06x = 1.6
0-7
‘l' 0-06" Z 1-6
0.9
0.06x =
0.06x
: 0.9
6x =
2 90
90
xX =
Z 15
15
monitor is
the monitor
width of the
The width
G!J
The
is ~x.
331. Since
Since the
the
8.
8'
length
the length
twice the
is twice
a rectangle
perimeter of a
perimeter
rectangle is
width ,
the Width,
twice the
plus
plus tw1ce
0
+ 5x
: 2(x
5)
2(x -_ 5)
Sx =
88 +
88 +sx
+ 5 x=: 22xx ‐- 110
0
i
1
2x+2<§x>
+ 2 ( X) = 4488
2x
3x =
= -‐18
18
3x
X
2
2x+gxz48
2x + x = 48
33
x
= 48
§x=48
~x
3
xzw-§=w
= 48 · -38 = 18
X
301
301
zw- 6
=
CHAPTER 30 ANSWERS
CHAPTER
ANSWERS TO THE
THE EXERCISES
9. I60 IConverting
9.
fractions in
in
Converting 75% and
and 85% to fractions
the equation
the
equation below,
below,
14. [E
equation to figure
u t x,
[QJMaking
Making an
an equation
figure oout
3xx-‐ 3 z
3
=2211
3
~(68)
= 17
17 n
20
17
17
51=‐ -20
En
51
n
4
3 x=
= 224
4
3x
x=
=8
2
_
8+1(8)‐12
20) _
51(fi
51 (~~) ‐
=n
n
the price
15. I90 IThree
Three times
times the
price of a shirt
shirt is 120.
Since a tie, which
costs 30, is k less
less than
that, k
which costs
than that,
must
30 = 90. As
As an
must be
be 120 ‐- 30
an equation,
equation,
60 =
zn
60
10.
10,
[z]
tie =
: 3(shirt)
3(shirt) -‐ kk
4 + NN _1
1
4+
3 0=: 33(40)
( 4 0 )-‐ k
30
15+
N _ 2
15
+ N
3 0=
: 1120
2 0-‐ k
30
Cross multiplying,
Cross
multiplying,
k == 990
0
2
( 4++N)
N )== 1
N
2(4
155++ N
8+ 2
N
2NN = 1155++ N
N = 77
Let the
board be
Then its
16. ~ Let
the width
width of the
the board
be w. Then
length
rectangle is its
length is 2w.
2w. Since
Since the
the area
area of a
a rectangle
length times
times its width,
length
width,
11. I48 IThey start
start with
with the
the same
same number
number x. Once
Once
gives 16to
16 to Julie,
Alice gives
with xx -‐ 16
16
Julie, Alice is left with
and Julie
and
Julie then
has x + 16.
then has
(2w)(w) =: 128
(2w)(w)
2
2w2=128
2w
= 128
w2=64
w2 = 64
xX ++ 116
6 = 22(x
( x ‐- 116)
6)
w=\/6_4=8
W
= v64 = 8
xX ++116
6=
=22xx ~- 3322
-‐ x == ‐-4488
12.
Finally,
2w =
= 2(8) =
: 16.
Finally, the
the length
length is 2w
z=448
8
x
X
17.
Cf]Let xx be
be the
the number
number of seashells
seashells that
that Carl
Carl
1
has. Bob then
has.
seashells and
has
then has
has éxx seashells
and Alex has
math is
Cf]The
The fraction
fraction of students
students who
who take
take math
1 1 11
11
11
11_Z
- - -‐ 5_
‐ -fl.Letxbethetotal
- - -5 =
. Let x be the tota 1
4 6 8
24
2
Since Alex and
~gxx seashells.
seashells. Since
and Bob together
together
number of students.
number
students.
have
have 60 seashells,
seashells,
11
1l X ‐=33
33
fix
24
1
3
-ixx +
+ 5x
- x ‐=60
60
2
2
z 72
=
72
2x =
: 60
2x
60
x
X
X
13.
[Q]Let x be
13. [E
Each trade,
trade,
be the
the number
number of trades.
trades . Each
[ a n has
Jason has
has aa
Ian
has a
a net
net gain
gain of 1 card
card while
while Jason
n
e t loss of 1 card.
card.
net
Carl
Carl has
has 30 seashells.
seashells.
2
= 44
4 4-‐ xX
200++Xx =
2
4
2xx =: 224
x
X
==112
2
302
=
= 30
30
PANDA
THE
COLLEGE PANDA
THE COLLEGE
Mark
books. Mark
number of books.
total number
the total
be the
Let x be
18. J 45 Let
J
;x
~x
Kevin has
and Kevin
then
has éx books
books..
books and
has ix books
then has
so
Mark, so
than Mark,
more than
owns 9 more
Kevin
Kevin owns
1
3
I
21. 40 Jessica runs at a rate of 4 yards per
takes for Jessica to
time it takes
the time
second.
be the
second . Let t be
equation
an equation
make an
can make
We can
overtake
overtake Yoona. We
and
distance
Yoona's distance and
being Yoona’s
with
side being
the left side
with the
distance.
the right
side being
Jessica’s distance.
being Jessica's
right side
J
1
4
= 99
- x ‐x -'‐ Ex
-3x
3
=44tt
300++ tt =
3
300= 33tt
1
100=s tt
by 12,
sides by
both sides
Multiplying
Multiplying both
4
: 108
4xx‐- 33xx =
catch up to
ItIt takes
seconds for Jessica to catch
takes 10 seconds
= 40
runs 4(10) =
time, Jessica runs
that time,
Yoona. In that
yards.
yards.
xX ==1108
08
Mark own
books is 108. Mark
number of books
total number
The total
ownss
22. 121 ILet the side length of the original patio
a length
patio has
renovated patio
Then the renovated
s. Then
be s.
be
has a
length of
an
up
Setting
5.
s + 4 and a width of s
s+4andawidthofs‐5.
Settingupan
patio,
renovated patio,
the renovated
area of the
the area
equation for the
equation
we get
owns
Kevin owns
and Kevin
i~ x 108 =
= 27books
27 books and
J
.
1
must then
Lon must
books. Lori
x 108 =
%x
z 36
36 books.
then own
own
3
books .
108 ‐- 27 ‐- 36 =
z 45 books.
out
given out
$5 coupons given
number of $5coupons
the number
[Q]Let
19. [E
Let the
given
$3 coupons given
number of $3coupons
the number
be ixt . Then
be
Then the
and the
out is 3x, and
out
the number
number of $1 coupons
coupons
out is 2(3x) =
given out
given
: 6x.
5)) = 990
((ss++ 4)(s
4 ) ( s- ‐ 5
0
52‐5‐20=90
s2 - s - 20 = 90
52‐5‐110=0
s2 - s - 110 = 0
( s-‐ 11
) ( s++ 10)
1 0 )== 0
ll )(s
(s
5(x) +
= 360
+ 3(3x) ++ 1(6x) =
5x+9x
= 360
6x =
+ 6x
5x + 9x +
positive
a positive
be a
must be
s must
length 5
side length
the side
Since the
original
the original
means the
which means
number,
number, ss = 11, which
2
sq ft.
(11) 2 =
= (11)2
was s52=
patio was
area
= 121 sq
area of the patio
20x =
= 360
x=
= 18
18
X
then
out is then
given out
coupons given
The number
The
number of $3 coupons
3x = 3(18) = 54.
can fill
itself can
Pipe A by itself
20. j 144 J Pipe
23.
rate
overall rate
the overall
done, r is the
work done,
amount of work
the amount
the
time
the
is
t
and
done, and
being done,
work is being
at
the time
which work
at which
the parking
finish the
spent. If Terry can
can finish
parking lot
lot in xat
spent.
Andy can
then Andy
himself, then
days
can finish it in 2x
days by himself,
paves
that Terry paves
means that
This means
himself . This
days
days by himself.
~ of the
the tank
tank
41-1
1
the
of the
fill Z,of
can fill
itself can
Pipe B by
and Pipe
hour , and
each hour,
each
by itself
6
can
used together,
When used
each hour
tank each
tank
hour.. When
together, they
they can
. 1 1 _ 55
fill
‐ 1‐2 of the tank
each hour
hour.. Now
Now
tank each
+ 8~ =
fill 7}+
12
Wisis 1
where W
formula W =
the formula
we
= rrt,
t , where
use the
can use
we can
5
hour, to find t.
per hour,
tank per
the tank
and r =
tank and
tank
: $5 of the
12
i
Andy
and Andy
day and
%of
each day
parking lot each
the parking
~ of the
X
day .
the parking
_.:!_
paves
of the
parking lot
lot each
each day.
paves 51;
2x
1
1
X
2X
- of the
+ 7x
pave -%+
Working
they pave
together, they
Working together,
5
1
‐ t
12t
1 == 12
12
12
rater r..
overall rate
the overall
day . This is the
each day.
parking
parking lot each
the
The
remaining fraction
fraction of the
the remaining
~ is the
number 2
The number
g
5 ‐= tt
takes
Therefore,
Therefore, itit takes
[II This problem follows the format of a
where W is
problem, where
word problem,
= rtrt word
standard
standard W =
This is
paved . This
be paved.
yet to be
has yet
that has
parking
parking lot that
given
bet.t. And
must
number 9 m
the number
W. 50
u s t be
A n d given
So the
the
be the
must
what
and r, t m
u s t be
Wand
have for W
we have
what we
and Andy
take Terry and
number
Andy to
days it will take
number of days
parking lot
the parking
remainder of the
pave
pave the remainder
working
together .
working together.
2
12
pipes to
hours for both
hours
both pipes
5
result to
this result
Converting this
tank. Converting
fill
the tank.
fill the
2
minutes,
get m =
= 132
~ x 60 =: 144.
minutes, we get
303
CHAPTER 30 ANSWERS
CHAPTER
ANSWERS TO THE EXERCISES
EXERCISES
Chapter13:
Chapter
Minimum& Maximum
MaximumWord Problems
Problems
13:Minimum
CHAPTEREXERCISE:
CHAPTER
EXERCISE:
1.
Katherine w
i l l need
(m Katherine
will
need a total
total of
6. [[] Working
would
Working at the
the slowest
slowest pace,
pace, Jason
Jason would
take 100 +
-;-6
take
6m
at the
:::::16.67
16.67 hours.
hours. Working
Working at
fastest
take 100
100 -;-8
+8=
: 12.5
fastest pace,
pace, he
he would
would take
hours.
answer choice
choice between
those
hours. The
The only
only answer
between those
numbers is 16.
ttwo
w o numbers
28
28 x 4 =
z 112 batteries.
that there
are 6
6
batteries. Given
Given that
there are
batteries in aa pack,
batteries
she will need
pack, she
need
18.67 packs.
112 -;-6
+ 6 x:::::18.67
that
packs. Since it’s
it's implied
implied that
packs only
only come
packs
come in whole
whole numbers,
numbers, she
she
round up to 19 packs
she
needs
needs to round
packs to ensure
ensure she
has
enough batteries
has enough
batteries..
2.
3.
7.
[I] Martha
needs 16
Martha needs
16‐-
2.5 = 13.5 more
more
.
13.5
ounces
ounces of glue,
glue, which
which amounts
z 7.7
7.7
amounts to g
1.
:::::
75
glue
sticks. Since glue
glue sticks
sticks can
glue sticks.
can only
only be
be
purchased
whole amounts,
purchased in whole
Martha must
must
amounts, Martha
purchase a
a minimum
minimum of 8 glue
purchase
glue sticks.
sticks.
6 r11D-<t-
16..ow=teeS"
1 6 m 29612115
29.6.mt:'
1 student
student
X --X --119.115
9Jf1' xlgunce’
1 _o.i.mce100
.mt:'
IOOmE
X ----
6>-r_ |- x
a
:::::28.4 students
students
Since it wouldn't
wouldn’t make
make sense
sense to have
have
four
-tenths of aa student,
four-tenths
that can
can be
student, the
the most
most that
be
accommodated is 28
accommodated
28 students.
students.
0
To get
get the
the minimum
minimum number
number of greeting
greeting
cards
the
shop
could have
cards the shop could
sold, we assume
have sold,
assume
that the
shop sold
sold as
as many
many gift boxes
that
the shop
as
boxes as
possible (400 gift boxes).
possible
boxes). Since each
each gift box
was sold
sold for $7, the store
was
store sold
sold
400 x $7
$2,800
$7 =
= $2,
800 worth
worth of gift boxes
boxes in this
scenario, which
scenario,
least
which means
means the
the store
store sold
sold at least
$8,000
$8, 000 -‐ $2,
800 =
= $5,
200 worth
$2,800
$5,200
worth of greeting
greeting
cards . Since each
cards.
each greeting
greeting card
card was
was sold
sold for
store could
could have
$5, the store
have sold
sold a minimum
minimum of
$5,200-;- $5 = 1,040 greeting cards to meet its
goal.
goal.
8. @] Giovanni made 0.15 (25)( 12) = $45
during lunch.
lunch. If he
during
tables during
during
he serves
serves x tables
dinner, he
he will make
dinner,
make an
additional 0.15(45)x
an additional
0.15 (45)x
dollars. Since the
dollars.
should be
be
the total
total for the
the day
day should
least $180,
$180,
at least
45
+ 0.15(45)x
2 180
45 +
0.15(45 )x 2::
0.15(45)x
2135
0.15 (45 )x ~
135
x
X
1,800
the salon
salon needs
needs at least
1,550 =
: 22.5 ➔
‐> 23
least ~
toolkits. To get
get the required
required number
toolkits.
number of nail
nail
buffers, the
buffers,
the salon
salon needs
needs at
at least
least
4,000
4
, OOO:::::
27 toolkits.
toolkits . Based
m 26.7 ➔
‐‐> 27
Based on these
these
150
numbers, a minimum
numbers,
minimum of 27
27 toolkits
toolkits must
must be
be
purchased
the salon
purchased for the
salon to receive
the
receive both
both the
required number of nail files and the required
number of nail
number
nail buffers.
buffers.
0
Two liters is equivalent to 2 x 33.8 = 67.6
ounces, which will fill 67.6 -;- 12 :::::5.63 plastic
cups.
cups can
be
cups. Soat
So at most,
most , 5 plastic
plastic cups
can be
completely
completely filled.
304
135
~ o.15(45)
0.15 (45)
x ~2
X
4. I27 J To get
required number
get the
the required
number of nail
nail files,
5.
[m
0
20
TI-IE
COLLEGE PANDA
THE COLLEGE
PANDA
9. [!!] Leta
Let a be
be the
the number
number of fish
fish Ashleigh
Ashleigh
caught
caught and
and nn be
be the
the number
number of fish Naomi
Naomi
caught.
Using these
caught. Using
these variables,
variables, we can
can set
set up
up aa
system
system of an
an equation
equation and
and an
an inequality:
inequality:
11. []] Because
Because we're
trying to maximize
the
we’re trying
maximize the
number of nighttime
assume that
that
nighttime bottles,
bottles, we
we assume
number
65 daytime bottles
bottles were
65
only 65daytime
only
were filled.
filled. The 65
daytime bottles
65 x 2 = 130
ounces
daytime
bottles used
used up
up 65
130 ounces
the active
active ingredient
ingredient and
and 65
of the
65 x 66 = 390
ounces of flavored
flavored syrup,
syrup, leaving
leaving
ounces
the active
active
385 -‐ 130
130 =
= 255 ounces
ounces of the
ounces of
ingredient and
ingredient
and 850 -‐ 390 = 460
460 ounces
flavored syrup
ounces of
flavored
syrup.. The
The remaining
remaining ounces
255
. ingredient
.
d'1ent are
active
mgre
are enough
enoug hf for
or ¥ =
active
: 85
85
a = 33nn-‐ 9
45
aa + n z2:45
The
The equation
equation allows
allows us
us to substitute
substitute for
for a in
the
the inequality:
inequality:
3
aa+r1245
+ n 2:45
(3n
- 9) + n 2: 45
(3n‐9)‐+‐n245
4n254
2: 54
4n
n213.5
n
2: 13.5
nighttime bottles,
bottles, and
the remaining
ounces
nighttime
and the
remaining ounces
460
flavored syrup
enough for g
of flavored
syrup are
are enough
5
=
92
= 92
nighttime bottles.
bottles. Based
Based on
nighttime
on these
these numbers
numbers,,
we're limited
remaining amount
amount of
we're
limited by the
the remaining
active ingredient,
ingredient, so
so the
the maximum
maximum number
number of
active
nighttime bottles
nighttime
bottles that
that can
can be
be filled is
is 85.
Since
Since it's
it’s implied
implied that
that fish are
are caught
caught in
whole
whole numbers,
numbers, the
the minimum
minimum possible
possible value
value
of n is 14.
14.
12. [Q]
Let's set
set up a
a system
as the
the
El Let’s
system withs
with s as
as the
the number
number of
number of short
tables and
and I as
number
short tables
long tables.
tables.
long
10. I125 IIf
If we
we let
let b be
be the
the number
number of black
black
pebbles,
pebbles, w
w be
be the
the number
number of white
white pebbles,
pebbles,
and
and j be
be the
the number
number of jade
jade pebbles,
pebbles, then
then
4 S++ 8/
8 l ==1168
68
4s
j > ~
g and
and w < 2b. Since j = 32,
32, the
the first
s + Il §::;3322
inequality
inequality becomes
becomes 32
32 > ~,
g, which
which simplifies
simplifies
Divide
both sides
sides of the
the equation
equation by 4 to get
Divide both
get
to 64
so the
64 > b,
b,so
the maximum
maximum possible
possible value
value of bb
is 63. Using
Using b = 63, the
the second
second inequality
inequality
becomes
becomes w
w < 2(63), which
which simplifies
simplifies to
w <
< 126. Based
Based on this
this result,
result, the
the maximum
maximum
possible
possible value
value of w
w is 125.
125. Now
N o w you
you might
might be
be
wondering
why we
wondering why
we used
used the
the maximum
maximum
in the second
possible
possible value
value of bbin
second inequality.
inequality.
Since
Since w
w is less
less than
than 2b, maximizing
maximizing w
w means
means
that we
have to maximize
that
we have
maximize b first.
21:= 42.
42. Isolating
Isolating I then
21 -‐ ~.
2.
s5+
+ 21
then gives
gives /1=
= 21
Substituting this
result into
into the
inequality, we
we
Substituting
this result
the inequality,
get
get
s
s + 21 - -2 <
32
-
:. < 11
2 -
s::; 22
Based on
this result,
the maximum
number of
Based
on this
result, the
maximum number
short
used is 22.
short tables
tables that
that can
can be
be used
get at least
$140 worth
worth of tacos,
tacos, a
13. [!!] To
To get
least $140
customer
would have
have to receive
customer would
receive at least
least
140
53.8 ➔
‐> 54 tacos
tacos (we
(we round
round up since
since
~m 53.8
~-!~
m
it’s implied
tacos are
are given
given in whole
it's
implied that
that tacos
whole
numbers
only). To receive
54 tacos,
numbers only).
receive at
at least
least 54
tacos,
the
would have
have to buy
buy at least
the customer
customer would
least
54
= 13.5 ➔
‐> 14
14burritos
(again, we
round up
up
=
burritos (again,
we round
4
since
implied that
are sold
sold in
since it’s
it's implied
that burritos
burritos are
whole
numbers). Therefore,
Therefore, 14
the
14 is the
whole numbers).
minimum..
minimum
54‐4
305
CHAPTER 30 ANSWERS TO THE EXERCISES
14.
[I] Let a be the number
the numbers
16. I54 ILet s, m, and
and Il represent
represent the
numbers of
of two-tier cakes and
be the number
b be
number of three-tier
cakes Ava
three-tier cakes
decorates . Using
decorates.
Using these
variables, we
we can
can set
set
these variables,
up the
the following
following system
up
system of inequalities:
inequalities:
small, medium,
medium, and
small,
boxes shipped,
and large
large boxes
shipped,
respectively. Based
respectively.
Based on
on the
the information
information given,
given,
50
ss + m
m + II==2 250
2
0 a+
+3
5 b~
§ 3360
60
20a
35b
s+m
lI >
>s+
a+b214
a
+ b 2'.14
Since m =
= 70, the
the system
system becomes
becomes
Note
converted 6 hours
360
Note that
that we converted
hours to 360
minutes to set
set up
minutes
inequality.. The
up the
the first inequality
The first
inequality
4a+
7b
5
inequality then
then simplifies
simplifies to 4a
+ 7b ~ 72. To
solve
this
solve this system,
system, we have
get the
the signs
signs
have to get
pointing in the
pointing
the same
same direction
so that
we can
direction so
that we
can
add the
add
Remember that
the inequalities.
inequalities. Remember
that
inequalities can
inequalities
can be
be added
added only
only if
if their
their signs
signs
point in the
the same
same direction
point
(do not
not subtract
subtract
direction (do
inequalities; think
inequalities;
only in terms
terms of adding
adding
think only
them) . Soif
So if we
we multiply
multiply the
them).
the second
second
inequality by -‐ 44,, we
inequality
up with
the
we end
end up
with the
following system
system (note
following
(note the
the sign
sign change):
change) :
80
ss + II ==1 180
0
Il >> ss ++770
IsolatingI
Isolating l in the
the equation
equation gives
gives Il =
= 180 ‐- 5.
s.
Substituting this
Substituting
inequality, we
we get
get
this into
into the
the inequality,
1 8 0- ‐ss > ss ++7700
180
110>25
110
> 2s
5
>s
555 >
Since
Since s5is
implied to
bea
whole number,
number, the
the
is implied
to be
a whole
greatest possible
greatest
possible value
value of s5 based
on this
based on
this
result is 54.
result
4a+7b
4a + 7b S
~ 72
72
-‐ 44aa -‐ 44b
bS
6
~ -‐ 556
17.
Adding the
Adding
16,
the inequalities,
inequalities, we
we get
get 3b
3b 5
~ 16,
which simplifies
which
5 5.33. Since
Since it's
simplifies to b ~
it's
implied
implied that
that Ava decorates
whole
decorates cakes
cakes in whole
numbers,
numbers, the
the maximum
maximum possible
possible value
value of b is
5. This question
could’ve also
been solved
solved
question could've
also been
through guess
through
guess and
and check.
check.
[zJBased
Based on
on the
the given
11=
10
given information,
information, n
= 10
and w =
and
= Bx
8x (8 ounces
ounces of water
water in each
each of the
the
cups) . To use
use these
x cups).
these values,
we first set
set up
values, we
the following
the
inequality:
following inequality:
CS
~ 16%
16%
100n <
100n
< 116
6
nn + w -
15. [RJ
E] For
For 11 pound
seasoning, Lianne
Lianne will
will
pound of seasoning,
need 0.75 pounds
need
salt and
pounds
pounds of sea salt
and 0.25 pounds
black pepper.
of black
The sea salt
pepper. The
salt will
will cost
cost
0.75 x $2
the
black
pepper
$2 = $1.50 and
and the black pepper will
will
cost 0.25 x $8
Altogether, that'
that’ss
$8 = $2. Altogether,
$1.50 +
+ $2 =
= $3.50 for each
each pound
pound of
seasoning . Since Lianne
seasoning.
can spend
more
Lianne can
spend no more
than $210, she
than
she is limited
limited to making
making
.
$210
: 60
Therefore,
$
=
60 pounds
pounds of seasoning.
seasoning . Therefore,
350
ssfso
60
maximum.
60 is the
the maximum.
100(10)
10 < 16
100
10 +
8x s- 16
i 8.3
1000 ~
S 16
16(10
8x)
1000
(10 + Bx)
1000 ~
S 160
160 +
+ 128x
128x
1000
840 S
~ 128x
6.56 5
~ x
X
Since the question
Since
question indicates
indicates that
that x is aa whole
whole
number,
number, the
the minimum
minimum possible
possible value
value of
of xx
based
to note
note
based on
on this
this result
result is 7. It’s
It's important
important to
that we were
thatwe
sides by
by
were able
able to
to multiply
multiply both
both sides
10+
sign
10 + 8x
Bx without
without worrying
worrying about
about a sign
change
because 10
10+
8x is guaranteed
be
change because
+ Bx
guaranteed to
to be
positive (x must
positive
must be
be positive
positive in the
the context
context of
the problem).
problem) .
the
306
THE COLLEGE
COLLEGE PANDA
THE
PANDA
Chapter 14:
14: Lines
CHAPTER EXERCISE
EXERCISE::
1.
1.
[filA vertical
line that
vertical line
that intersects
intersects the
the x-axis at
5.
3 has
has an
an equation
equation of xx =
z 3.
2.‑
2. [f]
of line/ goes up three units for
every ttwo
every
w o units
its
units to the
the right,
right , which
which means
means its
slope is ~.
2 A parallel
line must
must have
have the
slope
parallel line
the same
same
slope. Only
slope.
Only answer
answer choice
choice (C)
(C) gives
gives an
an
equation
slope.
equation of a
a line
line with
with the
the same
same slope.
- y1 z}1
yz‐y1
Y2
xX2z -‐ xX)1
n
l
7 1- ‐ 1
3
1
1)
=3
n -‐ 1l
--=
1
-
5 -(ng
6.
B
From the graph, we can see that
J
goes up
to the
up 11 unit
unit for every
every 2 units
units to
the right,
right,
6 =3
6
3
1
which
which means
means its slope
slope is ~.. Since g is
E
n -‐ 11 = 2
perpendicular to J,
f, the
the slope
slope of gg must
must be
perpendicular
be -‐22
(the
negative reciprocal).
g passes
(the negative
reciprocal). Since g
passes
n=3
3.
3.
[f] The graph
through
point (1,
we can
through the
the point
( 1, 2),
~), we
can use
use
0
Draw
the x-intercept
to
Draw aa line
line from the
x-intercept of -‐22 to
the y‐intercept
y-intercept of -‐ 44..
point-slope
equation of
point -slope form to find the equation
of g:
yy ‐- yy1
1 ==" m(x
1 ( x-‐ xxi)
i)
5
y‐§‐‐2(x‐1)
y --=2(x - 1)
yll
2
5
2x+2+2
yy == _- 2x+
2 + -_
2
9
y‐‐2x+§
y = - 2x + 2
.
‐4
Finally,g (- 1) = - 2(- 1) +
229 = 2 + 4~2l =
l
6!
2·
A quicker
been to work
quicker way would've
would've been
work
you can
see, it goes
units down
down for every
every
As you
can see,
goes 4 units
4
2 units
to the right.
The slope
slope is
is _74
units to
right. The
~ = -‐ 22..
backwards
backwards from
from the
the point
point (1,
( 1, g),
~), knowing
knowing
that
the slope
slope is -‐ 22.. So
Soon
graph of
g, 11
that the
on the
the graph
of g,
unit
brings us
us to
unit to the left brings
.
. 8 ‐ 5 ‐_ 9 ‐_
4.-Theslopeof1mells
4. 0 The slope of line / is 6‐‐(‐_3)
( ) = ~ = !3..
9
6- - 3
3
(0,2.
= (0,4%>,
( 0, ~ +
+ 2)
2) =
(0,41), and
and 11 more
more unit
unit to
to the
the
Using
Using point-slope
point‐slope form,
form,
left brings
brings usto
+2)
= ((‐1,6%).
us to ((‐1,4%
- 1,41 +
2) =
- 1,6D.
=
m(x
x1))
yy -‐ yY1
l =m
( x - 11
1
yy‐8‐§(x~6)
- 8 = (x - 6)
Therefore, g(
g(‐1)
Therefore,
- 1) = 6.5.
3
7. []]
1
yy = éxx +
+66
3
u = fl2 -=0 =i 2_ = _ 1l
xz‐xl
0 -‐ 4
5
-‐ 44
2
answer choice
choice by
At this
this point,
point, we
we test each
each answer
plugging
x-coordinate and
and verifying
verifying
plugging in the x-coordinate
the
y‐coordinate. Only
answer (A)
( A ) works.
they-coordinate.
Only answer
works.
8.
307
[filFrom the graph,
slope mis positive and
y-intercept
mb < 0.
y-intercept b is negative.
negative . Therefore,
Therefore, mb
0.
CHAPTER 30 ANSWERS TO THE EXERCISES
@The
= -‐ 22xx -‐ 2
9. [Q]
The line
line y =
2 has
has a
a slope
slope of
of -‐22
and a
a y‐intercept
and
u s t have
have a
y-intercept of -‐ 22.. Line
Line Il m
must
a
slope
reciprocal of
of -‐ 22,,
slope that
that is the
the negative
negative reciprocal
1
which
which is 2'. Since
Since they
they have
have the
the same
same
from
from the
the table.
table . Answer
Answer A
A works
works for
for Monday
Monday
(c(7.2) =
= 30(7.2)
+ 400 =
= 616) but
o t for
any
30(7.2) +
but n
not
for any
of the
days.
Answer
B
works
for
the other
other days. Answer
works for
Saturday
Saturday (c(8.5)
(c(8.5) := 60(8.5)
60(8.5) -++ 210
210 =
= 720)
720) but
but
n
o t for any
not
any of the
the other
other days.
days. Answer
Answer D does
does
not
value of c for
not give
give the
the correct
correct value
for any
any of the
the
given
given values
values of 5.
s. Only
Only answer
answer C
C gives
gives the
the
correct value
correct
value of c for each
each of the
the given
given values
values
s. These
These types
of 5.
types of questions
questions require
require you
you to
be thorough.
thorough . Don’t
be
just test
and
Don't just
test one
one case
case and
choose the
choose
the first thing
thing that
that ”works.”
"works." You have
have
evaluate all
all the
the answer
to evaluate
answer choices.
choices.
2
y-intercept,
of line
must be
be
y-intercept, the
the equation
equation of
line Il must
yyzix‐Z.
= 1 x - 2.
2
10. []] The line we're looking for must have a
1
slope that
that is
slope
is the
the negative
negative reciprocal
reciprocal of
of %,
,
2
which is ‐- 22..
which
13. [E
[Q] A line
line with
with a
a positive
ot
positive y‐intercept
y-intercept will
will nnot
cross the
the y-axis
y-axis at
at a negative
negative point.
cross
point. Therefore,
Therefore ,
when x is 0, y cannot
cannot be
when
negative, which
which
be negative,
makes (E)
makes
answer.
(E) the
the answer.
y = ‐2x
+b
- 2x+b
Plugging
Plugging in the
the point
(1,5),
point (1,5
),
~
5 =: ‐2(1)+
- 2(1) + bb
1.6 or 2 IFirst,
First, plug
the point
14. 1l.6
plug the
point (2,6)
(2, 6) into
into the
the
7=b
equation of the
the line
line so
so that
that we
equation
we can
can solve
solve for a:
a:
Now that
that we
we have
have b, the
Now
+ 7.
the line
line is y = -‐ 22xx +
7.
1
a(2)
a(2)‐§(6)=8
- (6) = 8
3
@
11. [Q]
2 a- ‐ 22 =z 8
2a
10
-4 _2
2
10‐4
= 3x ‐- l 1 _ 3
2
0
2aa =110
a=5
6 _z
2
6
x -1
‐ l ‐ 33
X
So the
the equation
equation of the
So
the line
line is 5x -‐
Cross multiplying,
Cross
multiplying,
which
which gives
gives x = g
~ == 1.6.
2 x-‐ 22 ==118
8
2x
2
0
2xx = 220
12.
8.
8.
The x-intercept
x-intercept always
The
y-coordinate of
of
always has
has aa y-coordinate
0, so if we
we plug
0,soif
we get
get 5x
= 8,
plug in Ofor y, we
5x =
8,
2
( x-‐ 11)) = 118
8
2(x
x
X
1
all
z
3y =
15.
==110
0
[1JOne
One easy
easy way
way to approach
approach this
this problem
problem is
is
make up
up numbers
to make
numbers for a and
and b.
b. Let
Let a = 1
and b = 2 so
and
that ~g = ;%.. Since
so that
Since the
the second
second line
line
[SJWe can
can use
t w o days
days
use the
the values
values from
from any
any two
find the
the line
line that
best models
to find
that best
Let’s
models the
the data.
data. Let's
use the
the values
values from
from Monday
use
Monday and
and Thursday
Thursday to
616 -‐ 584
.
calculate the
the slope:
slope: ---calculate
= 80. At this
this
=
7.2 -‐ 6.8
point, we
can tell
point,
we can
tell the
the answer
answer is probably
probably
going to be
be choice
choice C, but
going
find the
but let’s
let's find
the
y-intercept just
y-intercept
just to be
be sure.
sure. Currently,
we have
have
Currently, we
80s +
c=
z 805
-+- b. Plugging
= 7.2 and
= 616
Plugging in
inss =
and c
c=
from Monday,
Monday, we
from
: 80(7.2)
‐+‐ b,
we get
get 616 =
80(7.2) +
which gives
gives 1)z
which
b = 616 ‐- 80(7.2)
80(7.2) = 40.
Therefore, c(s)
c(s) = 80s
805 +
+ 40.
Therefore,
d
perpendicular to the
is perpendicular
the first,
first, C;=
- = -‐ 22,, which
which
e
satisfies the
the condition
satisfies
condition in answer
answer choice
choice (A).
An alternative
alternative solution
solution is to test
test each
each of the
the
answer choices
choices by
answer
in the
the values
values
by plugging
plugging in
308
THE
COLLEGE PANDA
THE COLLEGE
PANDA
Chapter15:Interpreting LinearModels
CHAPTEREXERCISE:
1. [I]
means the
decreases by 33 feet each
each day.
E The
The slope
slope is -‐ 33,, which
which means
the water
water level
level decreases
E The value
value 18
18refers
refers to the
the slope
slope of -‐ 18, which
which means
loaves remaining
18
means the number
number of loaves
remaining decreases
decreases by 18
2. []]
each
bakery sells
loaves each
each hour.
hour .
each hour.
hour. This implies
implies that
that the
the bakery
sells 18
18loaves
3.
[g They -intercept of 500 means that when n = 0 (when there were no videos on the site), there were 500
members.
members.
4. @ The number 2 refers to the slope of - 2, which means two fewer teaspoons of sugar should be added
(C) and
which
for every
Don 't be
fooled by answers
answers (C)
every teaspoon
teaspoon of honey
honey already
already in the
the beverage.
beverage. Don’t
be fooled
and (D), which
(hands,
slope is always
always the
change in y for each
"reverse"
”reverse” the x and
and they
the y (I:
and s, in this
this case).
case). The slope
the change
each unit
unit increase
increase
in x, not
n o t the other
other way
way around.
around.
5.
[I]
The salesperson earns a commission, but on what? The amount of money he or she brings in. To get
that
average price
that,, we
we must
m u s t multiply
multiply the
the number
number of cars
cars sold
sold by
by the average
price of
of each
each car. Since xx and
and c6 already
already
represent
cars sold,
respectively, the
represent the commission
commission rate
rate and
and the
the number
number of cars
sold, respectively,
the number
number 2,000 must
must represent
represent
the
the average
average price
price of each
each car.
6. [) The number 2,000 refers to the slope, which means a town 's estimated population increases by 2,000
for each
each additional
additional school
school in the
the town
town..
7.
[I]
8.
[fil When t =
9.
[I]
The number 4 refers to the slope of - 4, which means an increase of 1° C decreases the number of
words, the
the milk
hours faster.
hours
hours until
until aa gallon
gallon of milk
milk goes
goes sour
sour by 4. In other
other words,
milk goes
goes sour
sour 4 hours
0, ther e is no time left in the auction. The auction has finished. Therefore, the 900 is the
auction price
price of the
the lamp.
lamp.
final auction
Because it's the slope, the 1.30 can be thought of as the exchange rate, converting U.S. dollars into
euros. But after
after the
conversion, 1.50 is subtracted
subtracted away,
which means
means you
you get
get 1.50 euros
you
euros.
away, which
euros less than
than you
the conversion,
should have
have.. Therefore,
Therefore, the
best interpretation
interpretation of the
y-intercept is a
e u r o fee the
the bank
bank charges
charges
should
the best
the 1.50 y-intercept
a 1.50 euro
to do the conversion.
conversion.
. .
~
22
99
see the answer
answer more
more clearly,
clearly, we
we can
can put
the equation
equation mto
mtoy =
= mx
mx+
form: tt == 5xx + 5 . The
The slope
slope
10. L:.!JTo see
put the
+ bb form:
. To
5
5
2
is ~,
5, or 0.4, which
which means
means the
the load
load time
time increases
increases by 0.4 seconds
each image
image on the
the web
web page.
page.
seconds for each
The slope
slope is
is the
the change
change in y (daily
(daily profit)
profit) for each
(cakes sold).
11. @ The
each unit
unit change
change in xx (cakes
sold).
Notice that
the y-intercept
negative. It is
when no
no cakes
Therefore,
is the
the bakery’s
bakery 's profit
profit when
cakes are
are sold.
sold. Therefore,
12. [) Notice
that they
-intercept is negative.
anything that
that varies
varies with
with the
the number
number of cakes
cakes sold
sold is incorrect.
example, answer
answer (D) is wrong
anything
incorrect. For example,
wrong because
because
the cost of the cakes
cakes that
that didn
didn't
depends on
on how
many
It’s n
o t aa fixed number
number
' t sell depends
how man
y the
the bakery
bakery did
did sell. It's
not
like they
the y-intercept
The best
best interpretation
interpretation of the
the cost of running
running the
the bakery
bakery (rent
(rent,,
like
-intercept is. The
theyy‐intercept
-intercept is the
labor,
machinery,, etc.), which
which is likely
likely aa fixed number
number..
labor, machinery
13.
The solution
solution (5,0)
(5,0) means
means that
that the
the bakery's
bakery's daily
zero when
when 5 cakes are
are sold.
sold.
[J) The
daily profit
profit is zero
Therefore,,
Therefore
selling five cakes
cakes is enough
enough to break
break-even
with daily
selling
-even with
daily expenses.
expenses.
slope of the equation
equation is 5, which
which means
means the
temperature goes
up by 5 degrees
degrees every
every hour
hour.. So
50
the temperature
goes up
14. I2.s j The slope
every half
half hour
minutes), the
the temperature
goes up
= 2.5 degrees.
every
hour (30 minutes),
temperature goes
up by 0.5 x 5 =
degrees.
309
309
CHAPTER 30 ANSWERS TO THE EXERCISES
.
.
.
1
1x
mx
1
15. [E
[[I Putting
Putting the
the equation
equation m
t o y = mx +
+ b form,
form, y = 5x +
one more
more turtle
into
+ 7. The
The slope
slope of % means
means that
that one
turtle
requires
additional half
requires an
an additional
half aa gallon
gallon of water
water.. So
SoIll
11] is true.
true .
Getting x in terms
terms of y, x = 2y
Getting
The "slope”
water can
can support
support ttwo
wo
2y ‐- 14. The
"slope " of 2
2 means
means that
that 11 more
more gallon
gallon of
of water
more turtles.
turtles. SoI
more
So I is true.
true.
16.
CgBecause
Because this
this question
question is asking
asking for the
the change
change in "x"
” x ” per
y ” (the
reverse of
slope), we
per change
change in ”"y"
(the reverse
of slope),
we need
need
the equation
equation to get x in terms
to rearrange
rearrange the
C.
terms of C.
C = 1.5 +
+ 2.53:
2.Sx
Dividing each
Dividing
each element
element in the
the equation
by 2.5,
equation by
: 0.6 + x
0.4C =
= 0.4C -‐ 0.6
x =
The slope
slope here
here is 0.4, which
means the
The
which means
the weight
weight of aa shipment
shipment increases
increases by 0.4 pounds
pounds per
per dollar
dollar increase
increase
in the
the mailing
a 10
increase in the
mailing cost is equivalent
equivalent to a
mailing cost.
cost. So
50 a
10 dollar
dollar increase
the mailing
a weight
weight increase
increase of
10
pounds.
10 x 0.4 =
: 4 pounds.
310
THE COLLEGE
COLLEGE PANDA
THE
PANDA
Chapter
Chapter 16:
16: Functions
CHAPTER EXERCISE:
CHAPTER
EXERCISE:
l.1.
[Q]
FE] Check
Check each
each answer
answer choice
choice to see
see whether
whether
10.
J (O) == 20,J
f(0)
20,f(1)
and /f(3)
The only
only
(l ) = 21, and
(3) == 29. The
function that
that satisfies
satisfies all three
function
three is (D).
(D).
2. [Q]J
@f(x)
t w o graphs
graphs
(x) := g(x)
g (x) when
when the two
intersect.
They
intersect
intersect.
intersect at 3 points,
so there
points, so
there
must be
be 3 values
must
values of x where
x ) = g(x).
where Jf ((x)
conditions .
conditions.
11. lg(2)
[I)g( 2) =: 22
22 -‐ 11=
= 3.
113.So,
So,
2+1=
ff(g(2))
(g(2)) =
J
(3)
=
3
= f(3) = 32+1 2 110.
0.
(3) = -‐ 22.. N
3. @ Jf(3)
o w where
?
Now
where else
else is ffatat -‐ 22?
When x =
When
z ‐- 33.. Soa
be -‐ 33..
So a must
must be
4.
12. ~
B The
The difference
difference between
2x22 -‐ 2 and
between 2x
and
2x +
+ 4 is a constant
In other
constant of 6. 1n
other words, 6
needs to be
needs
= 2x
2x22 -‐ 2 to get
be added
added on to y =
2
y=
2x +
: 2x2
+ 4.
entails a translation
translation 6 units
units
4. That
That entails
upward.
upward.
[I] Draw
Draw aa horizontal
at y = 3.
3. This line
horizontal line at
line
intersects
intersects J[ ((x)
x ) four times,
there are
four
times, so there
are four
solutions
(four values
values of x for which
solutions (four
which
J (x ) =
f(X)
= 33).
)‑
5.
@ The x‐intercepts
x-intercepts of ‐3
- 3 and
and 2 mean
mean that
that
fJ((x
x ) must
and (x -‐ 2).
2).
must have
have factors of (x + 3) and
That
A y-intercept
y‐intercept of
That eliminates
eliminates (C) and
and (D). A
when we plug
= 0,
12 means
means that
that when
plug in x =
x) =
answer
(B)
meets
Jf ((x)
= 12. Only
Only answer
meets all these
these
[g Plug
Plug in -‐33 and
and 3 into
into each
each of the answer
answer
13.
whether you
choices to see whether
you get
get the
same
the same
value.
you're smart
smart about
about it,
realize
value. If
If you’re
it, you'll
you’ll realize
2 , which always
that answer
answer (C) has
has an
an xx2,
that
always gives
gives
a positive
positive value
a
value.. Testing
Testing (C)
(C) out,
out,
3(‐3)2
+11 = 28
and
Jf ((-‐ 33)) = 3(3)2 +
28 and
2
= 3(3)
3(3)2 +
+11 =
= 28. The answer
indeed
ff(3)
(3) =
answer is indeed
[I) Draw
Draw a
a horizontal
z c,
horizontal line
line at
at y =
C, passing
passing
through (0, c). This horizontal
horizontal line
through
line intersects
intersects
with f three
times. That
with
three times.
That means
means there
there are
are 3
values
values of x for which
x) =
= c.
which Jf ((x)
14.
14.@
(C).
(C)-
g(k)
= 8
8
g(
k) =
6. @] First, g(10) =
6.-First,g(10)
= f(20)
o w,
f (20) -‐ l.1.N
Now,
f(20)
Finally,
f (20) = 3(20) + 2 = 62. Finally,
g(10) = 62
62‐- 11 = 61.
_ 22
iol
16 + (- 4)
32
7.
‐ 44))=: 16‐2%4)i
: 3‐1
2
7. .~ f (J (2(- 4)
=
- 8 =
8.
Lookingat
only when
when
Looking at the chart,
chart, fJ (x) z
= 44 only
x=
=3.3. So
Sokk = 3.
-4 4..
15.
[I] We plug
plug in values
values to solve
solve for a and
and b.
lg]
g(x)
constant of 7 to
[Q]Since g(
x) just adds
adds a constant
every
the maximum
every value
value of f(x),
J (x), the
maximum of g(x)
must occur
occur at the same
must
x-value as
same x-value
as the
maximum
maximum of f(x),
J (x), namely
namely x = 3. So, the
maximum
reached at
the point
point
maximum of g(x) is reached
at the
(3,g(3)),
since g(x)
: Jf(x)
7, this
this
(3,g (3)), and
and since
g (x ) =
(x) + 7,
point is
is (3,f(3)
(3,/ (3) +
point
+ 7).
Plugging in (0,
(O, -‐ 22),
) , -‐22 =
: a(0)2 + b =
: b. So,
Plugging
b = -‐ 22.. Plugging
Plugging in (1,3),
(1, 3),
a(1)2+
3 = “(1)2
+b
3=
z a -‐ 2
= a
5=
16. .0 f ( J1(18)
8 ) =: \/18
= v'16
\/1‐ =
z 4.
4.
✓18 ‐- 2 =
=
/f(11)
(11) =
So a = 55 and
Soa
x) =
: 5x
5x22 -‐ 2. Finally,
Finally,
and Jf ((x)
\/11‐
/If=2
= \/§
3.
=
J9 == 3.
= 4 -‐ 3 =
Testingeach
ff(18)
(18) ‐- ff(11)
(11) =
= 1. Testing
each
[J((3)
3 ) == 5(3)2 ‐ 22 == 43.
5(3)2 -
9. 0
2f(k)
=8
2f
(k) =
8
flJ (k)
k) =
Z 44
J
answer
also
answer choice,
choice, f (3) is the only one
one that
that also
equals 1.
equals
l.
Plug in the
and check.
check.
Plug
the answer
answer choices
choices and
1
1 2 1
. .
1
f(§)
_ (5)
‐ z,whlchlslessthan
2'
J (1)=
(1)
=
~,which is less than 1
2
The answer
(A).
answer is (A).
311
CHAPTER 30 ANSWERS TO IBE EXERCISES
17. ~
factor g,
I If
If we
we factor
3, we
we get
get
2 + 4x + 4 = (x + 2) 2 . Since g(x) is
g(x)
=
x
g(x )= x + 4 x + 4 ‐_ (x+2)2. Sinceg(x) is
f (x)) shifted
shifted k units
units to the
the left,
25. [[I
each of the
the answer
answer choices
When
E Test each
choices.. When
x == -‐ 33,, Jf (x) = -‐22 according
according to its
graph ,
its graph,
g(x) = ((-‐ 3 ++3)(‐‐3
3)( - 3 -‐ 1)
0.Inthis
and g(x)
1) = 0.
In this
case
,
greater than
than g(x).
g(x ). When
case, f ((x
x ) is not
n o t greater
When
and
x == -‐ 22,, Jf ((x)
x) ~
z 1.5 and
g(x) =: ((-‐ 2 + 3
3)(
this case,
g(x)
) (-‐ 2 -‐ 1)
1) =
z -‐ 33.. Ilnn this
case,
fJ((x)
x ) >> g(x) so
so we have
have oour
u r answer
answer..
g(x)
+kk))
g ( x )== J(x
f(x+
(x+2)2=(x+k‐3)2
(x
+ 2)2 = (x + k - 3)2
+ 22 =: xx +
x+
+ k -‐ 3
2=
= k -‐ 3
5=k
26.
the value
value of x that
that makes
makes the
the substituted
the
substituted
equal to O.
0. This
value tells
tells you
you
expression equal
expression
This value
what
the horizontal
horizontal shift
For choice
what the
shift is. For
choice A,
Alternatively,
this
Alternatively, we
we could've
could’ve solved
solved this
and g(x)
que
stion by comparing
question
comparing J(x)
flat) and
g(x) to
to
2 . The
y=
offfJ((x)
- 3)
3
: xx2.
The graph
graph o
x ) == (x
( x~
3)22 is
i s3
2, and
units
units to the
the right
right of y =
: xx2,
and the
the graph
graph of
2.
g(
x) = (x +
the left
g(x)
+ 2)
2)22 is
is 2 unHs
units to
to the
left of y = xx2.
Therefore,
the left
left of J(
x ).
Therefore, g(x)
g(x) is 5 unit
unitss to the
f (x)
2
= 5 makes
makes 3x -‐ 2 equal
equal to 0, so
so the
x=
the
3
horizontal
units to the
right. For
For
horizontal shift
shift is ~g units
the right.
2
2
that value
the shift
shift is g
choice B, that
choice
value is -‐ g,, so
so the
3
the
expres sion 2x -‐ 33 equal
the shift
the expression
equal to 0, so
so the
shift is
3
~5 units
units to the
the right.
This is the
answer . For
right. This
the answer.
For
[Qj
@
g(
g(aa) = 6
choice D, the
the shift
shift is ~3 units
units to the
the left.
left.
choice
v13a
=6
36
3a = 36
27. [Q]
The key
are "linear
": f is
[E The
key words
words are
”linear function
function":
is
a
what straight
line can
a straight
straight line.
line. So
So for what
straight line
can
both
f(4)
2 /f(5)
be true?
both f(2)
/ (2) 5
'.Sf[(3)
(3) and
and J(
4) 2:
(5) be
true?
Only
Take aa minute
Only a
a horizontal
horizontal straight
straight line.
line . Take
minute
to think
since f is a
flat
think that
that through.
through . Now,
Now , since
a flat
line
10, then
all values
values of
are 10,
10,
line and
and f/ (6)
(6) = 10,
then all
offf are
no matter
value of x is. Therefore,
no
matter what
what the
the value
Therefore,
f mO)) =
: 110.
a
f(
12
a = 12
E] Using
the tabl
table,
g(‐1)
: 2. Then,
Then,
20. [Q]
Using the
e, g(
- 1) =
/(
2) == 66.
f (2)
21
21.
3
units
left. For
For choice
units to the
the left.
choice C, x = ~3 makes
makes
18. [Q]
2) cannot
E) (1,
(1,2)
cannot be
be on
on the
the graph
graph of y since
since an
an
x-value
of
1
would
result
in
division
by
0.
x-value
would result division
0.
19.
[g For
For horizontal
horizontal shifts,
the trick
trick is
to find
shifts, the
is to
find
[]Ju g(c) = 5, then
then c =
since 1 is the
the only
only
1 since
input that
gives an
an output
5. Then,
Then,
input
that gives
output of 5.
f(c)
f(C) == fJ( (l1 )) == 33.‑
28.
Remember that
8JRemember
that you
you can
can use
use your
your
calculator
for graphing
graphing.. The
The graph
graph of
calculator for
x ) = xx33 is ”centered"
(0,0).
Jf ((x)
"centered" at
at (0,0
). The
The graph
graph of
g(x)
is ”centered”
at (3,
(3, -‐2).
Comparing
g(x) is
"centered" at
2). Comparing
these
see that
that g(x)
g(x)
these points
points of reference,
reference, we
we can
can see
is shifted
shifted 3 units
to the
and 22 units
units
units to
the right
right and
downward
downward from
from [(x).
J(x) . Therefore,
Therefore,
g(x)
: J(x
f ( x -‐ 3) -‐ 2,
2,which
= -‐33
g(x) =
which means
means a =
From the
the second
second equation,
= 20. So,
So,
22.. []] From
equation , f (a) =
f(a)
a+
+ 55
J(
a) = -‐ 33a
20 == -‐3a
20
3a + 5
3a == -‐15
3a
15
a =z -‐ 5
and
um a +
+b
The ssum
bisis then
then
and b = -‐ 22.. The
= -‐ 55..
-‐ 33 +
+ ((-‐ 22)) =
.f-)g(
: 3=) /( 2(3)
2 ( : 3- )1)
‐ 1=) /(
f(5)=2.Weget
23. 8]g( (3)
5) = 2. We get
from the
the table.
table.
f (5) =‐2 2 from
f@)f(8)
= 4(8) -‐ 33=
Testing each
each
24. [Q]
J (8) =
= 29. Testing
answer choice
choice to see
see which
which one
one yields
yields 29, we
answer
that g(8)
3(8) = 3(8) + 5 = 29.
see that
see
312
PANDA
COLLEGE PANDA
THE COLLEGE
THE
9.
they-intercept
When x =
29. @ When
= 0, y =
= 9,
9, so
so the
y-intercept is 9.
3.
is
x-intercept
the x-intercept
so the
0, x = 3, so
When y = O,x
When
31.
slope of
with aa slope
line with
a line
function g(x) is a
The function
[g The
draw g(x)
If you
a y-intercept
and a
l1 and
y-intercept of k. If
you draw
g(x) with
with
the
from
k
for
possibilities
different
the
the different possibilities
from the
an
there's an
that there’s
see that
you'll see
choices, you’ll
answer
answer choices,
when
only when
with J(x)
points with
intersection
f (x) only
intersection of 3 points
below .
shown below.
as shown
k = 1 as
y
A
y
and
3 and
base of 3
a base
with a
triangle with
right triangle
a right
AAOB
6 AOB is a
theorem
a
pythagorean theorem,,
the pythagorean
Using the
height of 9. Using
a height
A
W2
0Bfi2z=AAB
+o
A0d2 +
¢
A §2
+ 3§2 == AB
92 +
s =zABfi2
90
3
¢ E =: AAB
B
3v'10
30.
32.
( - a, a),
Plugging in (‐a,a),
[I]Plugging
a=
(‐a
+ l12
2
a)) +
= aa(-
where J
units up
f
from where
up from
graph of g is 4 units
The graph
[g The
a:
- a 2 + 12
a = ‐a2+12
a2+a‐12=0
a2 + a - 12 = 0
and
the x and
slope off
the slope
is,
because the
of f is -‐ 22,, the
but because
is, but
y intercepts
will not
increase by
by the
the same
same
not increase
intercepts of g will
So
a ratio
amount.
increase in a
ratio of 2:1. 50
They'll increase
amount. They’ll
the
4, the
up by 4,
shifted up
when the
gets shifted
-in tercept gets
theyy-intercept
when
x-intercept gets
the right
right by
by 2. The
The
shifted to the
gets shifted
x-intercept
n
e w x-intercept
x-intencept is therefore
therefore 1I + 2 = 3.
new
solve for
Another
actually solve
this is to actually
do this
way to do
Another way
form,
-intercept form,
the
slope-intercept
Using slope
x-in tercept. Using
the x-intercept.
we
get
f(x)
=
‐
2
x
+
2.Adding
4
to
get
the
the
get
to
4
Adding
2.
2x
J(x)
get
we
equation
of
g,
g(x)
‐
2
x
+
6.
Setting
Setting
6.
+
2x
=
g(x)
equation
the
g(x) = 00 and
and solving
solving for x to get
get the
g(x)
x-intercept,
get x =
z 3.
we get
x-intercept, we
((a+
a + 44)(a
) ( a-~ 3
3)) == 0
a=
,3
= ‐-44,3
S
i n c ea a>>0,
0 ,aa=
= 33..
Since
313
ANSWERS TO THE EXERCISES
CHAPTER 30 ANSWERS
CHAPTER
EXERCISES
Quadratics
17:Quadratics
Chapter17:
Chapter
CHAPTER EXERCISE:
CHAPTER
1.
x-intercepts.
the x-intercepts.
[g We factor
factor to find the
6.
vertex. Since m is
occurs at its vertex.
always occurs
always
a
upwards in a
opens upwards
positive,
parabola opens
the parabola
positive, the
with
dealing
we're
means
which
shape,
”"U"
U ” shape, which means we're dealing with
Since
vertex . Since
its vertex.
at its
minimum at
the
parabola 's minimum
the parabola's
expanding
vertex form
form of
a vertex
us a
gives us
expanding f gives
f(x)
1]=
( x-‐ m)2 ‐- mm,,
m(x
=m
m [(x - m)2 - 1]
J (x) = m[(x‐m)2‐
the
Therefore, the
vertex is at (m, ‐- m). Therefore,
the vertex
the
m).
occurs at (m, ‐- m
parabola’s
).
minimum occurs
parabola 's minimum
3x - 10 = (x - 5) (x + 2 )
y=
y=x2‐3x‐10=(x‐5)(x+2)
x2 -
distance
The distance
and ‐- 22.. The
are 5 and
x-intercepts are
The x-intercepts
The
them is 5 ‐- ((-‐ 22)) =
between them
between
= 7.
formula,
quadratic formula,
2. [}] Using
Using the
the quadratic
X
=
✓ (4 ) 2 - 4(1)( 2)
-‐ 44 fl±: ./(4)?‐4(1)(2)
2 (1)
2(1)
_
=
_
v'8
-‐ 44±i \/§
the
into the
equation into
the first equation
7.
Substituting the
7. [}] Substituting
second,
second,
22
_‐4:t2\/§
_#- 4 ± 2v12
‐3
ex
+cx
x2 +
- 3==x2
2
:=-‐ 22i ±
\ / ..fi
§.
2
0
+ at + 33
x +ex+
0 = x2
will have
equations will
The
have ttwo
wo
system of equations
The system
two
has
above
equation above has t w o
the equation
solutions
if the
solutions if
have ttwo
above to have
equation above
the equation
For the
solutions. For
solutions.
wo
be
must be
4ae, must
b2 - 4ac,
discriminant, b2‐
the discriminant,
solutions, the
solutions,
positive.
positive.
3
.‑
3. []]
2a2‐7a+3:0
2a2 - 7a + 3 = 0
((2a
2 a- ‐l1)(a
)(a
0
- ‐3)3 )=: 0
c2‐
0
e2 - 4(1)(3) > o
remember
this , remember
factoring this,
trouble factoring
had trouble
you had
If you
quadratic
the quadratic
use the
always use
can always
you can
that you
that
c2
0
12 > o
e2 -‐ 12
1
.
0.5.
= g,, or 0.5.
1, a =
< 1,a
Smee a <
formula.
formula. Since
2 > 12
> 12
cC2
2
4.
4.
Expanding everything,
[iJExpandingeverything,
choices, only
answer choices,
the answer
each of the
Testing
only
Testing each
than 12
bigger than
value bigger
gives a value
(A), ‐- 44,, gives
answer (A),
answer
when
squared.
when squared.
(2x‐3)2
+5
(2x - 3)2 = 44xx +
4x2‐12x+9=4x+5
4x2 - 12x + 9 = 4x + 5
4x2-16x+4:o
4x 2 - 16x + 4 = 0
treat the
points , treat
intersection points,
the intersection
8. @ To find the
equations.
system of equations.
as a system
equations asa
ttwo
w o equations
second ,
the second,
into the
equation into
the first equation
Substituting the
Substituting
- 16
b
.
. b
- - 4- =
= ‐T
is - z
solutions IS_E
the solutions
sum of the
The
: 4.
The sum
4:(x+2)2‐5
4 = (x + 2) 2 - 5
= (x + 2) 2
99=(x+2)2
a
5.
5.
parabola
a parabola
maximum of a
minimum or maximum
The minimum
[]JThe
get
side to get
left side
the left
the 8 to the
Move the
lg!Move
i±S3 = x + 2
xX :=‐ -5 5,, 1
the
either use
can either
we can
Now,
l0x -‐ 8 = 0. N
+ 10x
3x +
o w, we
use the
this case,
formula or factor
quadratic formula
quadratic
factor.. In this
case, we'll
we’ll
=0
factoring: (x +
with
go w
i t h factoring:
+ 4)(3x ‐- 2) =
points
intersection points
the intersection
y-coordinates of the
The y‐coordinates
The
the ttwo
so the
equation), so
the first equation),
(from the
be 4 (from
must be
must
wo
(1, 4).
and (1,4).
(- 5, 4) and
are (‐5,4)
intersection are
points of intersection
points
b, b must be
~ . Since a > b,bmustbe
orr xx == §.Smcea
S
o ,xx== -‐ 44 o
So,
= 16.
( - 4) 2 =
= (‐4)2
b2 =
and 172
‐4
- 4 and
314
PANDA
COLLEGE PANDA
THE COLLEGE
THE
9.
the
(3, -‐ 88),
vertex is at (3,
the vertex
Because the
[g Because
) , the
14.
the
Because the
(C). Because
(A) or (C).
either (A)
must be either
answer must
answer
use
can use
we can
(1,0 ), we
through (1,0),
passes through
parabola passes
parabola
potential
our
test oout
to test
point to
that point
that
ut o
u r ttwo
w o potential
(C), we
=
x
in
plug
we
When
answers.
answers. When
plug
= l1 into
into (C),
(C).
answer is (C).
the answer
that the
0, confirming that
get y = 0,confirming
get
0
the
into the
equation into
Substitute
the first equation
Substitute the
second,
second,
- 3 = ax2 + 4x - 4
‐3=ax2+4x‐4
0=ax2+4x‐1
0 = ax2 + 4x - 1
the
solution, the
real solution,
one real
have one
For the
system to have
the system
real
one
only
have
should
above
equation
equation above should have only one real
discriminant,
the discriminant,
words, the
other words,
solution . In other
solution.
equal 0.
must equal
b2 - 4ac, must
b2‐
t(S -‐ t),
t2 =
v = St
equation 0
the equation
10. ) 2.5 ) From
From the
5t -‐ t2
: t(5
Oand
are 0
the t-intercepts
that the
can see
we can
we
see that
t-intercepts are
and 5.
5.
occurs at the
maximum occurs
the maximum
Because the
Because
the vertex,
vertex,
average of the
the average
t-coordinate is the
whose t-coordinate
whose
the two
two
maximum
the
in
results
2.5
=
t
t-intercepts,
t-intercepts, t =
results the maximum
graphing
by graphing
this by
confirm this
can confirm
v. You can
value of 1).
value
calculator .
your calculator.
on your
the
equation on
the equation
2
0
4 (a)( - 1) =
(4)
(4)2 -‐4(a)(‐1)
=0
1 6++ 4a
4 a== 00
16
4
6
4aa== ‐- 116
I
number of
minimum number
the minimum
11. ) 400 To find the
that it
so that
must
company m
the company
mattresses the
mattresses
u s t sell
sell so
money, set
lose money,
doesn't lose
doesn’t
set P =
= 0.
a=
= ‐- 4
15.
m2
120,000 =
= 0o
m2 -‐ 100m -‐ 120,000
Since
value of J
minimum value
the minimum
results
f (x). Since
results in the
opens
that
parabola
a
offf is aparabola that opens
the
graph of
the graph
offf
minimum of
the minimum
shape, the
a ”"U"
upwards
U ” shape,
upwards in a
located at
which is located
the vertex,
occurs
vertex, which
occurs at the
xx =_-_baE=_- _- (24it) = 12.
12. Therefore,
Therefore, the
the
2 1
2
2a
2(l)
units each
produce 12
should produce
manufacturer
12 units
each
manufacturer should
unit.
per unit.
cost per
the cost
minimize the
week
week to minimize
(m -‐400)(m
+ 300) = 00
400)( m +300)
m=
= ‐300,400
- 300,400
number of
the number
sense for the
make sense
doesn't make
Since it doesn’t
Since
negative , m =
sold to be negative,
mattresses
: 400. If
If
mattresses sold
above
equation above
the equation
factoring the
trouble factoring
had trouble
you had
you
and
calculator and
your calculator
on your
tough), graphing
(it's tough),
(it’s
graphing on
good
both good
are both
quadratic formula
the
formula are
the quadratic
alternatives..
alternatives
12.
that
value of x that
the value
looking for the
We're looking
[ill We're
02]
are x-intercepts,
x-intercepts ,
bare
and x = b
0 and
Since xx = O
16. El Since
and
Oat
f (x), the
at x = 0 and
speed, is 0
transfer speed,
data transfer
the data
transfer
x=
: b. First,
would the
data transfer
the data
why would
First, why
transfer is
the file transfer
O? Well, the
at x =
Oat
be 0
speed
: 0?
speed be
just
starting
so
no
megabytes
have
been
just starting so no megabytes have been
the
number 10,000
The number
10,000 is they-intercept,
the y-intercept, the
[g The
number
the number
total
when x, the
expenses when
monthly expenses
total monthly
expenses
of tables,
is
0.
We
can
assume
these
expenses
these
assume
can
tables,
etc.
salaries, etc.
worker salaries,
equipment, worker
rent, equipment,
be rent,
to be
the speed
transferred
yet. Now
why would
would the
speed
Now why
transferred yet.
the file
that the
n S W e r iis
s that
best aanswer
The best
= bb?? The
transfer
no
are no
there are
so there
completed, so
just completed,
has just
transfer has
transfer - just
to transfer‐just
more
left to
data left
megabytes of data
more megabytes
at
stops at
when it stops
Owhen
like
be 0
would be
speed would
car's speed
like a car's
likely
most likely
Therefore, b most
trip. Therefore,
a trip.
end of a
the end
the
represents
which the
transfer
the file transfer
time at which
the time
represents the
completed.
completed.
b
a t xx =
bee 0Oat
square. First
First
the square.
complete the
need to complete
We need
13. [[] We
divide
by -‐ 1,
everything by
divide everything
y = x2 - 6x - 20
-‐y=x2‐6x‐20
term by 2 to
the middle
divide the
Now
N
o w divide
middle term
to get
get -‐33
the -‐33
put
and
square
that
result
to
get
9.
We
put
the
get
result
that
and square
subtract the
and subtract
with x and
parentheses with
the parentheses
inside the
inside
the
9 at
end.
the end.
at the
17.
and
0 and
g(x)
and x =
=O
parabola and
x) is aa parabola
Since g(
[g Since
x=
= ~g is the
the
x-intercepts, x =
its x-intercepts,
are its
= c are
= (x‐3)2‐20‐9
(X - 3)2- 20 - 9
-‐ y =
parabola’s
along
line along
the line
symmetry, the
axis of symmetry,
parabola's axis
vertex
the vertex
case, the
this case,
vertex lies. In this
the vertex
which the
which
the
since the
occurs, since
is where
maximum occurs,
the maximum
where the
parabola
an
downwards in an
opens downwards
parabola opens
back
everything back
N
o w simplify
multiply everything
and multiply
simplify and
Now
by
1.
by -‐1.
y: ‐(x‐3)2+29
upside-down
U ” shape.
%is
the
is the
Therefore, ~
shape . Therefore,
upside-down “"U"
time
which the
a s at a
was
speed w
transfer speed
data transfer
the data
time at which
maximum
maximum..
315
CHAPTER 30 ANSWERS TO THE EXERCISES
18.
[Q]One of the x-intercepts
is 3. Since the
the
at the
must lie at
5, must
vertex, 5,
the vertex,
x-coordinate of the
x-coordinate
other
the other
x-intercepts, the
the ttwo
midpoint of the
midpoint
w o x-intercepts,
us
giving us
7, giving
Therefore , k = 7,
x-intercept is 7. Therefore,
x-intercept
the
plug in the
now
can n
We can
3)( x -‐ 7). We
= a(x -‐ 3)(x
y=
o w plug
solve for a.
point to solve
as a point
vertex as
vertex
‐- 332
2 = a(5 -‐ 3)(
3)(55 -‐ 7)
a(2)( - 2)
‐- 332
2 = a(2)(‐2)
‐- 332
2 := ‐- 44a
a
a
a == 8B
19.
both
into both
(3, k) into
point (3,
the point
Substituting the
[I] Substituting
equations,
equations,
k :=22(3)
( 3 ) +b
k=(3)2+3b+5
k = (3)2 + 3b + 5
Substituting the
equations. Substituting
system of equations.
This is a system
This
the
second,
the second,
into the
equation into
first equation
2
( 3+
) +bb=: (3)2+3b+5
(3)2 + 3b + 5
2(3)
6+
+ b := 9 +
+ 3b
3b + 5
+bb 2=33b
6+
b ++114
4
: 22b
b
-‐ ‐88=
: ‐- 4
b=
equation,
first equation,
the first
From the
From
k = 6+
+ bb == 66 -‐ 4 = 2
316
THE
THE COLLEGE
COLLEGE PANDA
PANDA
Chapter 18: Synthetic
Chapter
Synthetic Division
Division
CHAPTER
CHAPTER EXERCISE:
EXERCISE:
[g
1.
l.
5. [TIzz ‐- 11is
only ifif the
5.
is aa factor
factor only
the polynomial
polynomial
yields O
yields
0 when
when z2 =
= 1 (the
(the remainder
remainder theorem).
theorem).
Therefore,
can set
set up
up an
an equation.
equation.
Therefore, we
we can
4
2 I4x
x ‐ 2
4x
4x
4 x -‐ 8 8
2(1)3‐kx(1)2+5x(1)+2x‐2=0
2(1) 3 - kx (1) 2 + 5x (l ) + 2x - 2 = 0
2
‐kx+5x+2x‐2=O
2 - kx + 5x + 2x - 2 = 0
X -
l
8
3
+ 7x
7 x:= 0
-‐ kkxx +
0
From
can see
see that
that k =
From here,
here, we
we can
2 7.
7'
This
This result
result can
can be
be expressed
expressed as
as4
+ _‐xfB_2
4+
x- 2
z-
‘
7'‑
7. []]
3x
3 x ++ 1
2 x++11I6x
6 2x 2+ +5x
5 x++ 22
2x
6x 2
6x
6. [2J
By the
the remainder
6.
E By
remainder theorem,
the remainder
theorem , the
remainder
2
is(‐4)2+2(‐4)+1=16‐8+l=9.
is (- 4) + 2(- 4) + 1 = 16 - 8 + 1 = 9.
‘
xX
+
+ 33xx
‐
-
3X2 _ 21
3x 2 - 2x
2x + 11
‐ 6x ‐ 4
6x
11
This
This result
result can
c a n be
be expressed
as
expressed as
.
1
3x + 1
1 + -m,
Wthh Q
Q=
= 3x
3x +
+ 1.
1.
, from
from which
2x + l
-‐
,
expression by 2x -‐ 1l and
expression
and write
write the
the result
result in
in
the
form of
the form
of
Dividend
: Quotient
Divisor + Remainder.
Dividend =
Quotient xx Divisor
+ Remainder.
2x +
+ 11
2x
2 -l-5
2 X- ‐ 11j 4x
4X
2x
+
!iD
‘
,
2x
.
'
_
_
. _
8.
to divide
diVide the
the
8. El This
This question
question 15
is asking
asking you
you to
expression
by xx +
+ 11 and
and write
the
result
in
the
expression by
write the result in the
form
form of
Dividend
: Quotient
+ Remainder
Remainder..
Dividend =
Quotient x><Divisor
Divisor +
6
6
Therefore,
Therefore, 4x
4x22 +
+5=
= (2x +
+1)(2x
1) +
+ 66..
1) (2x -‐ 1)
R2
= 66..
2x -
6
xx ++ 11j 2x
2 2x 2- ‐4x
4 x- ‐ 33
2
2x2
+ 2x
2x
2x +
2x +
2x
+ 55
2x
2x -_ 11
4.
4.
6x
+ 4
6x +
4
‐ 8
8
2 x ‐ 6
r
4x‐‘2x
-
4
This
This result
result can
can be
be expressed
expressed as
as
8
..
2 -‐ 3x
x _ , from
from which
xx -‐ 2
Whmh A
A=
_ xx -_ 2.
2'
3 _ 2'
2
@]
[E] This
This question
question is asking
you to
to divide
divide the
the
asking you
2
4x 2
2
3x‐23x2
x -‐4 4
3x
- 2 j 3x 2 ‐ 8Bx
2x
2x +
+ 2
3.
3.
2
1
.
-‐
6x ‐
3
3
-‐
6x ‐
6x
6
6
3
3
Therefore, 2x
Therefore,
2x22 -‐ 44xx -‐ 3 = (2x -‐ 6)(x
6)(x +
+ 11)) +
+ 3.
3.
[I] Using
Using the
theorem, the
the
the remainder
remainder theorem,
remainder
remainder when
g(x) is divided
divided by x +
+ 33 is
is
when g(x)
equal
equal tog
to g ((-‐ 33)) =
= 2.
2.
317
317
CHAPTER 30 ANSWERS TO THE EXERCISES
9. ~ This question is asking you to divide the
expression by x ‐- 22 and
expression
in the
and write
write the
the result
result in
the
form of
form
Dividend =
= Quotient
Quotient ><
Dividend
Remainder,
x Divisor
Divisor + Remainder,
where ax + b
where
is the
is the
the
bis
the quotient
quotient and
and c
c is
remainder
remainder..
14.
be
be a factor
factor of p(x) ifif p
+ 6
+
2
xX -‐ 221xx 2 + 4x
4 x- - 99
x
X
+
xx22 -‐ 22x
x
6x x - ‐ 9
6
6xx -‐ 112
6
2
3
Therefore, x2 + 4x - 9 = (x
Therefore,x2+4x‐9=
( x ++ 66)() (xx-‐ 22)) +
+ 33..
Finally,
a=
F
i n a l l y,
a =1,l ,bb== 6,
6 ,cc=
=3
a n daa++bb++ cc ==110.
0.
3,,and
10. ~ Using
Using the
the remainder
remainder theorem,
theorem, p(2) =
=0
means that
that x -‐ 2 is a
a factor
factor of p(x).
means
11. ~ Use
the remainder
remainder theorem
theorem to test
Use the
test each
each
option for a
a remainder
remainder of 0.
option
23 + 22 - 5(2) + 3 =
p(2) =
=23+22‐5(2)+3
= 55..
13 + 12 - 5(1) + 3 = 0.
p(1) =
: 13+12‐5(1)+3=0.
p(- 3) = (‐3)3
(- 3)3 +
(- 3)2 -‐ 5(-3)
5(- 3) +
p(‐3)
+ (‐3)2
+ 3 =z o.
0.
Therefore,
divisible by x -‐ 1 and
Therefore, p(x) is divisible
and x + 3.
12.
[ill ItIf p(x) is divisible
divisible by x -‐
[E]
2, then
then p(2)
must equal
equal 0 (the
(the remainder
remainder theorem).
must
theorem).
Testing each
each answer
answer choice,
choice, only
Testing
only choice
choice (D)
results in 0 when
results
: 2.
when x =
13. ~ Using
Using the
the remainder
remainder theorem,
theorem, we
we can
can set
set
up a
a system
up
system of equations
equations.. When
When the
the
polynomial is divided
divided by
by x ‐- 1or
polynomial
1,the
1 or x + 1,
the
remainder
is
0,
which
remainder
which means
means that
that if we
we let
let
denote the
the polynomial,
polynomial, p(1) = 00 and
p(x) denote
and
p(- 1)== 00.
p(‐1)
a(1)4 + b(1) 3 - 3(1)2 + 5(1)
a(1)4+b(1)3‐3(1)2+5(1)
=o
=0
{ a(‐1)4
a(- 1)4 +
b(- 1)3 -‐ 3(‐1)2
+ 5(5(‐1)
=o
+ b(‐1)3
3(- 1)2 +
1) =
0
a+
+ b ‐- 3 +
+ 5
[I] From
From the
the remainder
remainder theorem,
theorem, 3x -‐
= 0
=
{ a ‐- b ‐- 3 ‐- 5 z= 0
Adding the
the equations
equations together,
Adding
together,
2 a- ‐ 66=: 0
2a
a= 3
318
(g)
(1)= 0.0.
11 must
must
THE
PANDA
COLLEGE PANDA
THE COLLEGE
Chapter19:Complex Numbers
CHAPTER EXERCISE:
.I(5‐3i)‐(‐2+5i)=5‐3i+2‐5i=7‐8i
= 5 - 3i + 2 - Si = 7 - Bi
i (ii(i++11)) == ii22++i =i :-‐1l + ii
2. I[[j
1. [g (5 - 3i) - (- 2 + Si)
.Ii4+3i2+2=1‐3+2:0
+ 3i + 2 = 1 - 3 + 2 = 0
5 ( ‐i)i )++ 6(1)
6 ( 1 =) :44-‐ 22ii
0 2 + 3i + 4i + 5i + 6i = 22 ++33ii ++44(-( ‐ 11))++ 5(4...2+3i+4i2+5i3+6i4
2
3. [g i4
2
3
4
S
n daa++bb=
: 22..
Sooaa=: 44,, bb == ‐-22,, aand
0
(6 + 2i)( 2 + Si) = 12 + 30i + 4i + 10i2 = 12 + 34i + 10(- 1) = 2 + 34i
5.. (6+21)(2+5i)=12+30i+4i+10i2=12+34i+10(‐1)=2+34i
=22..
Therefore,a
Therefore, a 2
lli i
Bii =
: 33ii ++66 ‐- 110
0+8
= -‐ 44 ++11
[g 3(i + 2) - 2(5 - 4i) =
6..@3(i+2)‐2(5‐4i)
[[l 3i(i + 2) - i(i - 1) = 3i2 + 6i - i2 + i = -‐ 3 ++ 66ii-‐ ((-‐ 11))++ ii = ‐- 22++77ii
7..I3i(i+2)‐i(i‐1)=3i2+6i‐i2+i=
(1)23. i = i
@]i93 = (i4)23. i = (1)23-i=i
8..Ei93=(i4)23-i:
0 (3 - i) 2 = 32 - 6i + ;2 = 9 - 6i - 1 = 8 - 6i
.-(3‐i)2=32‐6i+i2=9‐6i‐1=8‐6i
9.
10.
i2 - i4 =
0 Deal with the exponents first: (- i)2 - (- i)4 = 1'2‐i4
1Z'Dealwiththeexponentsfirst:(‐i)2‐(‐i)4
= ‐-
11.
(5 - 2i)( 4 - 3i) = 20 - 15i - Bi+ 6i2 = 20 - 23i - 6 = 14 - 23i
[[j
E(5‐2i)(4‐3i)zzo‐151‐81+6i2:20‐231‐6=14‐23i
1 -‐ 11=: -‐22
1
1
1
1
= ! - 1+1 = !
+ ..!_
!1 + ..!_
.'Z|7+i‐2+i‐4_7‐1+1_7
G\l
i
i
i4
i2
~ i
12
12.
~ = ‐i- i
= 1L=
getygz
I
byy i ttoo get~·~
Now
bottom b
and bottom
top and
both top
multiply both
Now multiply
I
.(1-3i)(3‐1’)_3‐i‐9i‐1~3i2
I
_ ‐ -1lOi
3_
3 -‐ 1lOi
3 - lOi +
3
0 i-‐ 3
0 i -2_ ‐- 1/.
!Al (1 - 3i) (3 - i) _ 3 - i - 9i + 3;2 _ 3‐101'+3i2
- -- ---,----,-- ----=--- 2
- -----=· -13 L.:2J-2
10
9- i
9 - 3i + 3i - i
(3 - i)
.
((33+
+ ii)) (3‐i)_9‐3i+3i‐i2:
9-12
“ 9 9‐ -( ‐(-11))
10
_
14
.
‐
3i2 _
_
‐
_
2
2
.(2‐i).(2‐i)_22‐2i‐2i+i2_4‐4i+i2_4‐4i‐1_3~4i
(2 - i) ' (2 - i) = 2 - 2i - 2i + i2 = 4 - 4i + i = 4 - 4i - 1 = 3 - 4i =
0
2
2
+ 22i i-‐ i 2 _
((22++ ii)) ((22- ‐ i)
i ) _ 44 ‐- 22ii +
4- i
4‐12
1) ) "
‘ 44‐- ( (‐- 1
5
5
!i
~ -_ 2
§
“5 5 5
5
IThe
i)(1 + i).
i).
denominator is (1 -‐ i)(1
common denominator
15.. [[] The common
4ii++ i 22) )++(2
((44++ ii)(
) (11++ ii)) (2
( 2-‐ ii)(
) (11-‐ ii)) _ (4
( 4++ii ++ 4
( 2- ‐ 2i
2 i-‐ ii++i 2i2 )) _ 44 + Si
5 i- ‐ 1l ++ 2 -‐ 33ii -‐ 1 _ 44 ++ 22ii
2
2==
11++ ii -‐ ii ‐- i i2
((1l ‐- i )i)(1
( 1 + ii)+(
) (ll ‐- i i)(1
) ( 1++i)i )=‐
‘
1 -(‐ ( ‐ 11))
'
2
: 22 ++ ii
=
319
CHAPTER
30 ANSWERS
ANSWERS TO
TO THE
THE EXERCISES
EXERCISES
CHAPTER 30
Chapter
Chapter20: Absolute Value
Value
CHAPTER
CHAPTEREXERCISE:
EXERCISE:
1.‑
1. []]
1|f(1)|
/ (l ) I =
1- 2(1) 2 - 3 (1) + 11 = l- 4 1 = 4
=l‐2(1)2‐3(1)+1|:|_4|=4
2.
The best
to solve
question is
is trial
and error
error.. If
= 1,
1, which
which is
is not
not
2. ~ The
best way
way to
solve this
this question
trial and
If x
x := 3,
3, for
for example,
example, 1|2
2 -‐ 33|J =
greater than
result indicates
that we
we should
should try
try larger
we continue
continue to
o u r way
greater
than 5. This
This result
indicates that
larger numbers.
numbers . If
If we
to work
work our
way
up,
arrive at
at the
minimum possible
possible value
value xx =
= 8,
which results
results in
in J2
|2 -‐ BJ
8| =
: 6.
6.
up, we
we would
would arrive
the minimum
8, which
3.
can equal
equal -‐55 (when
[]J Only
Only the
the expression
expression in answer
answer (B) can
(when x
= 11 or
Because the
absolute value
value of
=
or 3). Because
the absolute
of
anything
is
always
greater
than
or
equal
to
O,
the
other
answer
choices
can
never
reach
‐
5
.
anything is always greater than or equal to 0, the other answer choices can never reach - 5.
Recall that
that the
graph of y = Jx
[x]l is aa V-shape
V‐shape centered
centered at
The graph
graph pictured
pictured is
4. []] Recall
the graph
at the
the origin.
origin. The
is also
also
V‐shaped but
but converges
converges at
at y =
= -‐ 22,, which
which means
means it has
has shifted
w o units
V-shaped
shifted ttwo
units down.
down. Therefore,
Therefore, the
the equation
equation
of the
the graph
: lx
le l -‐ 2. Note
Note that
|x -‐ 21
2| shifts
shifts the
w o units
units to the
O T ttwo
w o units
units
graph is y =
that y = Ix
the graph
graph ttwo
the right,
right , N
NOT
down
down..
5.
5. [ill
IE Test each
choices, making
making sure
sure to include
negative possibilities.
possibilities. For
For example,
the
each of the
the answer
answer choices,
include the
the negative
example , the
answer is not
answer
n o t (A) because
z 2 or -‐ 2,
2, Ix
[x -‐ 33]J is n
o t greater
than 10. However,
However, Ix
[x -‐ 33]1is greater
greater
because when
when x =
not
greater than
than 10
when x =
than
10 when
: -~8.
8.
6.
@]Smart
Smart trial
trial and
and error
error is the
the fastest
fastest way
way to find
find the
the bounds
The lower
lower bound
for xxisis -‐88 and
and the
bounds for x. The
bound for
the
upper
upper bound
bound is -‐ 44.. There
There are
are 5 integers
integers between
between -‐44 and
do this
and -‐88 (inclusive).
(inclusive). If we wanted
wanted to do
this problem
problem
more
more mathematically,
mathematically, we
we could
could set
set up
up the
the following
following equation:
equation:
<
-‐3 3 <
+
<
x -+- 6
X
6< 3
Subtracting
6,
Subtracting 6,
< Xx << -‐ 3
-‐ 99 <
Since
Since xx is an
an integer,
integer,
-‐ 88 S
~ Xx ~g -‐ 4
7. [ill
El In the
the graph
graph of 1|f/ (x)I,
(x)|, all
all points
with negative
negative y-values
the x-axis)
x‐axis) are
across the
the
points with
y-values (below
(below the
are flipped
flipped across
x-axis. All points
points with
positive
y-values
with positive y-values stay
stay the
the same
same.. Graph
shows this
correctly..
Graph (D)
(D) is the
the one
one that
that shows
this correctly
8.
8.
positive,
[J] If nn is positive,
: 110
0
n -‐ 22 =
n
=
1
2
n = 12
IIff n iiss negative,
negative,
n -~ 22=-: ‐ 110
0
n
11
== -‐ 8
The
The sum
s u m of these
these ttwo
w o possible
values of 11
n is 12
12+
= 4.
possible values
+ ( -‐ 88)) =
320
THE COLLEGE
COLLEGE PANDA
THE
PANDA
9.
II] Make
Make up
up a
Let’s say
say x =
= 3.
3. Then
Then b =
= 13
|3 ‐a number
number for x. Let's
10]
7, and
= 4.
ur
101 =
= 7,
and b
b -‐ x =
= 77 -‐ 33 =
4. Using
Using o
our
numbers, we’re
numbers,
looking for an
answer choice
choice that
that gives
b
=
7.
The
only
one
that
does
so
is
(C).
we're looking
an answer
gives 4 when
when b = 7. The only one that does so is (C).
do this
To do
this question
mathematically, we
we have
have to
always negative.
question mathematically,
to realize
realize that
that when
when x < 10,
10, x -‐ 10
10 is
is always
negative .
Therefore,
Therefore,
x -‐ 10
10 == -‐bb
x z= 10
lO ‐- b
Using substitution,
Using
substitution, b -‐ x becomes
b) =
becomes b -‐ (10 -‐ b)
= 2b
2b ‐- 10.
10.
.
.
3.
II] The
The midpoint
6!41 and
midpomt of 61
62 is
15the
the average:
average: (6;
//22 =
= 6;
The midpoint
midpoint is
%away
from the
and 6~
(6!4 ++ 62)
6~)
6!.
The
is !
away from
the
4
4
2
4
boundaries
the accepted
boundaries of the
accepted range
length of a
m u s t be
be within
within !
%of
range for the
the length
a hot
hot dog.
dog. So
So whatever
whatever h
h is,
is, it must
of the
the
4
midpoint:
midpoint:
1
_ _
162‘<
}
[Q]
11. El Smart
Smart trial
trial and
and error
error is the
way to find
find the
lower bound
is -‐22 and
the fastest
fastest way
the bounds
bounds for n. The
The lower
bound for
for n is
and the
the
upper bound
bound is 6.
6. There
There are
are 9 integers
integers between
upper
between -‐22 and
and 6 (inclusive).
(inclusive). If
If we
we wanted
wanted to do
do this
this problem
problem
more
we could
more mathematically,
mathematically, we
set up
up the
the following
equation:
could set
following equation:
‐5 < n ‐- 2 < 5
-5
Adding
Adding 2,
-‐33 <
< n << 7
Since n
Since
71is an
an integer,
integer,
-~22 g
:S n 5:S 6
12. E]
[Q]The
The midpoint
midpoint of 400 and
and 410 is
is the
the average:
average: ((400
+ 410)
410) /2
/2=
z 405. The
The midpoint
midpoint is
400 +
is 55 away
away from
from the
the
boundaries
the accepted
boundaries of the
accepted range
range for the
the length
length of a
m u s t be
be within
a roll
roll of tape.
tape. So
So whatever
whatever Il is,
is, it must
within 5 of
the
midpoint:
the midpoint:
ll|l ‐- 405|
405 1 < 5
5
13. @ There
There are
are ttwo
w o possible
possible values
values of x,
x, 33 and
and -‐ 11.. There
w o possible
of y, 11 and
get
There are
are ttwo
possible values
values of
and -‐ 55.. We
We get
the
smallest possible
possible value
the smallest
value of xy when
when x =
= 33 and
and y =
z -‐ 55,, in which
which case
= -‐ 115.
5.
case xy =
14.
II] If |a|
1, then
laI << 1,
then by definition,
definition,
‐1
- 1 < a < l1
This means
that III
true . Because
This
means that
I I I is true.
Because a must
m u s t be
be aa fraction,
fraction, a2
1,so
also true.
true. However,
o t always
a 2 < 1,
so II is also
However, I is n
not
always
1
true
when a is negative,
true because
because when
5 is
is not
n o t greater
greater than
negati ve, !
than 1.
1.
a
..
..
3
1 ..
3
1
.
. i 1
1o7 The m1dpomt
1 away
15. ~
1 3 and
and 25
2 1 1s
The midpomt
of 11
is the
the average:
average: ( 1113 +
+ 21
= 2. The
midpomt is
15Z
away from
from the
the
2 1) //22 =
The midpoint
4
4
4
4
4
i
1
boundaries of the
the accepted
range for the
boundaries
accepted range
the weight
weight of a
is, it must
a muffin.
muffin . So
So whatever
whatever m
m is,
must be
be within
within 1 of
of the
the
midpoint:
midpoint:
1
Im
lm-_ 221
| << _-4
4
321
CHAPTER
CHAPTER 30 ANSWERS
ANSWERS TO THE
THE EXERCISES
EXERCISES
Chapter21:
Chapter
21: Angles
CHAPTER EXERCISE:
CHAPTER
EXERCISE:
@]
1. E] Using
Using the exterior
exterior angle
angle theorem,
theorem,
Because alternate
alternate interior
are
5. [}] Because
interior angles
angles are
equal,
the missing
equal, one
one of the
missing angles
angles of the
the lower
lower
triangle is also
also a:
triangle
a:
+ jj
kk = ii +
1 4 0=: 50
5 0++j j
140
9 0==jj
90
2.
triangle is
2. [}] The missing
missing angle
angle in the
the left triangle
is
180°
180° -‐ 60° -‐ 50° = 70°
70°.. This angle
angle is an
an
exterior
the right
exterior angle
angle to the
the triangle
triangle on the
right.. So,
using
using the
the exterior
exterior angle
angle theorem
theorem,,
7 0== y ++440
0
70
3 0=: y
30
an exterior
exterior angle
angle to the
Since x is an
the lower
lower
triangle,
triangle,
3. []]
equal to
sum of
the
.a a + b +
+ c + d is
isequal
to the
thesum
ofthe
angles
as shown
angles of the
the quadrilateral,
quadrilateral, as
shown below
below..
x = aa +
x=
+ bb
angle at
the triangle
6. []] The angle
at the
the top
top of the
triangle is
180
70 -‐ 30
we look
at the
larger
180 -‐ 70
30 =
= 80.
80. If we
look at
the larger
angle gives
triangle,
taking away
triangle, taking
away the
the top
top angle
gives
a + b.
= 180
180 -‐
+b=
a+
a
7.
80 = 100
100
form a circle, which
which means
[f] The angles
angles form
means
they sum
360°.
°.
they
sum to 360
xX +
=360°‐45°‐80°
=235°
+y =
360 ° - 45 ° - 80° =
235°
Because
angles of a
a quadrilateral
Because the angles
quadrilateral sum
sum to
to
360, the answer
answer is 360.
360,
4 0. ‑
8.
4.
Filling out
the bottom
the
[fJ Filling
out the
bottom triangle,
triangle, the
missing
angle is 180°
missing angle
180° -‐ 60° -‐ 40° = 80°,
which
the angle
across from
from it in the
the
which means
means the
angle across
upper
triangle is also
80°. Finally,
Finally,
upper triangle
also 80°.
x+
x ++yy))==1 180
80
+yy ++( (x
4
0
+
y
+
(
4
0
+
y
)
=
1
80
40 + + (40 y) = 180
2y+80=180
+ 80 = 180
2y
2yz100
= 100
2y
X
2
2180°
5 ° -‐ 880°
0 ° == 55°
z=
180° -‐ 445°
O
y z=S50
The ttwo
w o angles
angles form
form aa line,
line, which
which means
9. [fil The
means
they sum
sum to 180°
180°..
((xx + 4400))+
+ xX ==1180
80
2 x+ 4400 z=1180
80
2x
2x=
40
2x
=1140
xX ==7700
322
THE COLLEGE
COLLEGE PANDA
THE
PANDA
10.
15. -Angleais
I260 IAngle a is equal
equal to
60 = 120.
to180 -‐ 60
Angle b
bisis equal
equal to 180 -‐ 40
Angle
40 = 140. Finally,
Finally,
a+b=120+140
=260.
a+ b = 120 + 140 =
260.
[Q]We can figure out two angles within
the
triangle:
triangle: 100°
100° and
and 50°. Because
Because yy is an
an
exterior
the exterior
exterior angle,
angle, we can
can use
use the
exterior angle
angle
theorem
theorem to get
get its
its value:
value:
yy:100+50=150
= 100 + 50 = 150
n . III
Shaded
Shaded Angles
Angles = Angles
Angles of Rectangle
Rectangle
+ Angles
Angles of Quadrilateral
Quadrilateral
= 360 +
+ 360
=
= 720
=
polygon sum
12. @]
E] The
The angles
angles of any
any polygon
s u m to
180(n -‐‐ 2), where
n
is
the
number
where
the number of sides.
sides.
The angles
of
a
hexagon
angles a hexagon (6 sides)
sides) sum
s u m to
180(6
180(6 -‐ 2) =
= 720. Because the
the hexagon
hexagon is
regular,
regular, all angles
angles have
have the same
same measure
measure..
Therefore,
Therefore, each
each angle
angle is 720 7+ 6 =
= 120°
120°..
FinalJy,
Finally,
X
x = 120 -‐ 90 = 30
30
13.
[Q]
[Elb is an
an alternate
alternate interior
interior angle
angle to the
the 45°
angle,
they 're equal:
equal: bb =
angle, which
which means
means they’re
= 45. aa
and c are
are also
also alternate
alternate interior
interior angles
angles so
so
and
z c = 180 -‐‐ 45
45 = 135. Using
Using these
values,,
a=
these values
can see that
that all
all three
three are
are true
true..
we can
14.
two missing
missing angles
angles in the
the smaller
smaller
IIIThe two
triangle add
add up
up to 80°. The two
bottom
triangle
two bottom
angles in the larger
larger triangle
triangle add
add up
up to
angles
70 =
= 110. If
If we
we take the
the two
t w o missing
missing
180 -‐ 70
angles
smaller triangle
away from
from the
the
angles of the smaller
triangle away
two
bottom
angles
of
the
larger
triangle,
we’ll
two bottom angles
larger triangle, we 'll
end up
with x +
+ y.
end
up with
x + yy ==11
0 ==330
0
x+
ll00 -‐ 880
i,
r.
I
I
I
t_
\
I
"'·
)
I.
'
\
323
323
THE EXERCISES
ANSWERS TO THE
CHAPTER 30 ANSWERS
CHAPTER
EXERCISES
Triangles
Chapter 22: Triangles
EXERCISE:
CHAPTER EXERCISE:
CHAPTER
1.
1.
always the
hypotenuse is always
Because the hypotenuse
[f] Because
7.
hypotenuse
be the hypotenuse
must
side, x +
largest side,
largest
+ 5m
u s t be
Using the
be the legs. Using
must
x and x ‐- 2 m
while atand
while
u s t be
pythagorean
theorem,
pythagorean theorem,
0
The side
v'4:= 2.
square is fl
length of the square
side length
create ttwo
triangle to create
Draw
wo
height of the triangle
Draw the height
30‐60‐90
triangles :
30-60-90 triangles:
x2+(x‐2)2:(x+5)2
relationsrup,
triangle relationship,
30--60-90 triangle
Using the 30‐60‐90
2. @ Using
1
1
1
1
(10) =
DC
_ 53C
_ E(10)
_ 5.
BC =
DC =
2
2
relationslup,
the 45-45-90
Using the
3. [f] Using
3.
45‐45‐90 triangle
triangle relationship,
xX = 6\/2.
6y1.
4.
then
triangle is then
the triangle
The area of the
1
similar .
are similar.
DCE are
and DCE
ABE and
Triangles ABE
[I]Triangles
(2)( ./3) =
1§(2)(\/§)
: v'3
\/5
Therefore,
Therefore,
CD
CD
AB
AB
6
3
AB
AB
be x.
piece be
bottom piece
the bottom
height of the
the height
8. @ Let the
x.
radii of the
the radii
and the
cone and
the cone
height of the
The height
shown
as shown
triangles as
similar triangles
form ttwo
circles form
w o similar
below.
below.
5CE == BE
E
5 =Z 4T
AB
8 = AB
5..
measure,
same measure,
the same
have the
angles have
w o angles
Iss IIfIf ttwo
I?
same
the same
have the
them have
opposite them
sides opposite
then the sides
then
we
perimeter, we
largest perimeter,
length. To get the largest
length.
perimeter
be 20. The perimeter
side to be
third side
choose the third
choose
is then
= 55.
then 15 + 20 + 20 =
6
.‑
6. lli]
Using
similarity,
Using the similarity,
1
2
-
1+X
6
Cross
multiplying,
Cross multiplying,
M
4
4
2x = 6
22 ++ 2x
0
Drawing
into ttwo
wo
base into
the base
splits the
height splits
the height
Drawing the
equal
parts of length
From the 3-4-5
3‐4‐5
length 4. From
equal parts
pythagorean
height is 3.
know the height
triple, we know
pythagorean triple,
xx 2
=2
The area is
: 12.
~ (8)(3) =
then %(8)(3)
is then
324
THE COLLEGE
PANDA
COLLEGE PANDA
9. I2.5 ITriangles
GHC are
similar.
Triangles GEF and
and GHC
are similar.
Solving for EF,
Solving
02]
13. [E We
We can
can use
use the
the pythagorean
pythagorean theorem
theorem to
to
find
find BC:
Ac2
: 13C2
AC 2 +
+ A32
AB 2 =
BC2
HC
EF
EG
EZFTG
HG
1222 + 9922 =
= BC
BC22
12
EF
EF_10
10
T2 ‘ 5?
225 := BC2
BC 2
E
EFF== 4
= BC
15 =
Note that
Note
that this
this is
is a
a multiple
multiple of the
the 3‐4‐5
3-4-5
triangle.
triangle.
Triangles ADF
Triangles
A D P and
similar. So,
and GEF are
are also
also similar.
AD_ D _GE
A
§
C
D
F
OFF _ EEF
AD
2
A_D_z
=
5
5 ‘4 4
5
AD‐§_2.5
AD =
2 = 2.5
10.‑
[g
10.
0 X _!!_
7T _57t
= 547T
x180°_
225°
225
180°
A
4
Now
N
o w ACDE
.6.CDE is similar
similar to ACAB.
.6.CAB .
The sides
= 1.5
1.5
11. [f] The
sides of triangle
triangle DEF
DEF are
are 9
9+6
6=
times longer
times
of
longer than
than the
the respective
respective sides
sides
triangle
triangle ABC.
ABC. Therefore,
Therefore, EF = 9 x 1.5 = 13.5
13.5
and
= 5 xx 1.5 =
= 7.5. The
of
and DF
OF =
The perimeter
perimeter of
triangle
+ 13.5 +
: 30.
triangle DEF
DEF is then
then 9 +
+ 7.5 =
12.
B
9
CB
2CE_ CB
DE
DE ‐ AB
AB
g
CE_ 15
15
6 '
shortest side
side in the
the isosceles
[g IfIf W
BC is the
the shortest
isosceles
9
Cross
Cross multiplying,
multiplying,
triangle,
AB == AC
triangle, then
then AB
AC and
and [A
L A is
is the
the
smallest
we
w a n t to
to
smallest angle.
angle . At the
the same
same time,
time,
want
maximize L
AA
so
that
AB
is
minimized.
N
o
maximize
A so that L B minimized . Noww
the triangle
if all the
the angles
angles were
were 60°, then
then the
triangle
would be
would
beequilateral
the
equilateral and
and EE
BC wouldn’t
wouldn't be
be the
shortest
So we need
need to decrease
decrease L
[AA to
shortest side.
side . So
the
minimizes
the next
next highest
highest option,
option, 50°, which
which minimizes
£3
= 65°.
L B to 130 + 2 =
6)
9(CE) =
: (15)(
(15>(6)
CE = 10
10
below and
14. []] Draw
Draw the
the extra
extra lines
lines shown
shown below
and
use
triangle.
use the
the 8‐15‐17
8-15-17 right
right triangle.
B
W
w
65°
8
17
1--------20
65°
CL---------~A
z
15
12
12
50°
X
325
15
y
CHAPTER 30 ANSWERS
CHAPTER
THE EXERCISES
ANSWERS TO THE
EXERCISES
15.
the
[f] Draw
Draw an
an extra
extra line
line to
to complete
complete the
19. a
[fJThe
The smaller
smaller triangle
triangle in
in the
the first
first quadrant
quadrant
is a
is similar
similar to
a 3‐4‐5
3-4-5 triangle
triangle and
and is
to triangle
triangle
AOB.
AOB. Using
Using the
the similarity,
similarity,
rectangle . Then use the 7-24- 25 right triangle.
28
OB _ 3
3
@
=-
15
15 ‘ 5
24
24
24
24
OB == 9
Therefore,
= -‐ 99..
Therefore, n =
28
7
20.
2
+2
+2
+7+
2
244+
288+
255+
+2288 ==11
112
16. ~ Using
Using the
the pythagorean
pythagorean theorem,
theorem,
0
The radii
radii extending
extending to
to the
the comers
comers of
of the
the
triangle split
triangle
split the
the circle into
into three
three equal
equal parts,
parts,
so
the measure
measure of angle
so the
ADB is
angle ADB
+ 3 = 120°. In radians,
360 +
radians , this
this is
o
n
200
TT: _ 271'
2n:
120
x 17305
_ ?.
l
X 180° = 3·
2
82+x2
=(x+2)2
8 + x2 =
(x + 2) 2
21.
64+x2=x2+4x+4
64 + x 2 = x 2 + 4x + 4
6
=4
644=
4xx ++ 4
6 0=
: 44xx
60
[fJ Because
Because triangle
triangle ABC
ABC is 45‐45‐90,
45-45-90,
AB = 2\/2.
2/2 . Because
AB
triangle ABO
ABD is
Because triangle
is
2
30‐60‐90,
30--60-90, AD = ii;;
\/§
1 5== Xx
15
17.
=
DB _
[fJ Label
Label what
know..
what you
you know
B
2 XX 1 C
2
1
Y
W
W
1
2
2
2ELl~
A
11ZZ
2
D
22.
All triangles
triangles in the diagram
45‐45‐90,
diagram are
are 45-45-90,
which means
which
: XY =
= Jz‘
means wz
WZ =
/2 and
and
WX = zr
ZY = N2.
wx
2/2. The
The perimeter
perimeter of wxrz
WXYZ is
then
fi + + fi/2+ +
2 \2/2
/ § ++22/2
\ / § :=6 6/2.
\/§.
then /2
18.
and
and DB is twice
twice that:
that:
4
4_fl
/2
$3
v'3
We can
rationalize the
We
can rationalize
the fraction
fraction by multiplying
multiplying
both the top
top and
and bottom
bottom by $3:
both
y'3:
zflxéz‘h/E
D B fi fi s
[i]
Because
Because the
the triangles
triangles are
are 45‐45‐90,
45-45-90,
BC =
is half
BC:
= i.~. The
The radius
radius of the
the circle
circle is
half BC:
\/§
( ~) (é)
( ~) 2
= L2
~.
(é)
[i]
From
= 7 and
From the
the coordinates,
coordinates, AB =
and
Because [LAABC
BC = 7. Because
B C is a right
right angle,
angle,
triangle A
triangle
ABC
45‐45‐90 triangle.
BC is aa 45-45-90
triangle .
Therefore,
Therefore, the
the measure
ABAC
= 45°,
measure of L
BAC =
Finally, the area of the
Finally,
the area of the
circle is
2
which is 45° x _!!_
which
182° =
g radians.
= !:
radians .
180°
4
326
PANDA
THE
COLLEGE PANDA
THE COLLEGE
23.
a
on a
triangle lies
lies on
equilateral triangle
the equilateral
Because the
[Q]Because
25. ED]
equal,
are equal,
sides are
their sides
side
square, all their
the square,
side of the
isosceles.
are isosceles.
DCE
6D
and A
6 ABE
means AA
which
BE and
C E are
which means
have 36°
both have
and 6
6 XYZ and
Outer AXYZ
w WXZ
a 2 both
GI]Outer
they are
and share
angles
angles and
share L.Z.
42. Therefore,
Therefore, they
are
the angles
Label the
similar. Label
similar.
angles with
with tick marks
marks ifif
similarity more
this similarity
see this
want to see
you
more clearly.
clearly.
you want
XZ
with W
correspond with
6 WXZ correspond
XZ in AWXZ
and W
W
WZ and
XYZ,
6X
YZ in A
and
Y Z , respectively
respectively (sides
(sides
and W
each
correspond with
angles correspond
the 36° angles
opposite the
opposite
with each
sides opposite
and sides
other
opposite the largest
largest angles
angles
other and
So,
other) . 50,
each other).
with each
correspond with
correspond
B
C
A
D
WXZ
from 6
W2
AWXZ
6 WXZ _ XZ from
from AWXZ
WZ from
6 XYZ
from AXYZ
YZ from
XYZ _ yz
from 6
XZ from
xz
AXYZ
xz
wz ‐ k,
..
WZ
XZ
Smce
equal to k,
be equal
also be
must also
XZ = k, fiYZ must
Smee E
24.
equal to
which
XZ (the
reciprocal) is equal
(the reciprocal)
~~
means XE
which means
[ A E D=
2 60°, L.BAE
[ B A E = L.CDE
ZCDE =
: 30°, which
which
L.AED
means
means
75° .
[L.ABE
A B E =
= é
A E B == ADCE
= [L.DEC
D E C =
: 75°.
L.DCE =
L.AEB
1
1
k'
k.
Finally,
Finally,
ZBEC
150°..
L.BEC = 360° ‐- 75° -‐ 75° -‐ 60° = 150°
Let AD
[[I Let
I
I
OD,
is parallel
parallel to fii
AC is
Since TC
26. 3.75 Since
L.BOO.
~ ZBDO.
L.BCA E
and ZBCA
[L.BAC
B A Cg
B O D and
~ [L.B00
similar.
6 BOD are similar.
and ABOD_are
6 BBAC
Therefore,
A C and
Therefore, A
OD are
and W
that AC and
OD = 8 and
Notice
and that
are
that CD
Notice that
imply
corresponding
given lengths
lengths imply
The given
sides. The
corresponding sides.
a 30-60-90
ADE is a
Because ADE
30‐60‐90
= x. Because
that
AE = 2x. Note
and AE
DE =
triangle, DE
triangle,
= xy13
x 3 and
Note that
are all
DCF are
BEF, and
6 ADE, 6
AADE,
ABEF,
and 6
ADCF
all congruent.
congruent.
B
the sides
that
sides of
are~2 the
6 BAC are
sides of ABAC
the sides
that the
will
BO will
length of 35
ABOD.
o w finding
the length
finding the
Now
6 BOD. N
BC.
length of W.
allow
the length
determine the
us to determine
allow us
point B down
Draw
vertical line
from point
down to the
the
line from
Draw a vertical
BO as
x-axis to form
with 35
as the
the
triangle with
right triangle
a right
form a
hypotenuse.
triangle is
this triangle
base of this
The base
hypotenuse . The
(they-coordinate
8 ‐- 2 z= 6 and
height is 8 (the y-coordinate
its height
and its
(a
triangle (a
of point
6-8-10 triangle
This is aa 6‐8‐10
point B). This
didn't
multiple of the
3‐4~5 triangle).
you didn’t
If you
triangle) . If
the 3~5
multiple
the
used the
know
also used
could've also
you could’ve
this, you
know this,
pythagorean
BD. Since
theorem to find BD.
pythagorean theorem
3
BD
: 10, BC=
gum) =
é(10) == 3.75.
= ~(10)
BC = ~(BD)
BO =
2x
A
xX
0
D
C
triangle ABC
The side
side length
ABC is 3x.
outer triangle
length of outer
The
DEF is xy13.
triangle DEF
inner triangle
length of inner
The
side length
x\/3.
The side
similar, the
are similar,
triangles are
the ttwo
Because
w o triangles
the ratio
ratio
Because the
the
square
the
equal
of their
areas
is
equal
to
the
square
of
the
areas
their
ratio
sides:
their sides:
ratio of their
Area
ofA
E F _ (x/3)
(x\/3)22 __ 3x
3x22 _a 1
1
DEF
6D
Area of
2
2
3
9x
(3x)
6 ABC _ (3x)2 " 9x2 _
Area
Area of AABC
327
CHAPTE
R 30 ANSWERS
ANSWERS TO THE
CHAPTER
THE EXERCISES
27. ~
IE]Draw
Draw a
a straight
straight line
line down
down the middle.
middle .
The length
length of this line
top
The
line is 9 because
because the
the top
part
simply a radius
part is simply
radius of the
the semicircle,
semicircle,
whose length
length is half
half the side
side of the
whose
square,
the square,
29. ~
y
6 --c+ 2 = 3.
A
Ah
3
B
Draw the extra line shown above to form a
30-60-90
30‐60‐90 triangle
are in aa ratio
triangle (the
(the sides
sides are
ratio of
11:: x/3
v'3:: 2). The
segment
The acute
acute angle
angle the
the line
line segment
forms with
forms
with the
the x-axis
x-axjs is 30°, which
which makes
makes
0
=
9 z 360 -‐ 30
: 330°. In
in radians,
radians, this
this is
30 =
H
_
117r
3300
330° X
x _!!__ = 117r_
180° ‘ “ 6Z “
180°
Using
Using the
the pythagorean
pythagorean theorem,
theorem,
¥
92 + 3§2 =
= AAB§2
%
A §2
90 = AB
V90= AB
JT=AB
3JE=AB
3M = AB
Because DBCE
DBCE is a square,
square, DB = 3 and
and
30. j 3.75 j Because
triangles ABD
ABO and
and DEO
DEO are
are simjlar
triangles
similar (their
(their
angles are
are the
angles
the pythagorean
pythagorean
the same).
same) . Using
Using the
theorem, D0
theorem,
5. Using
similarity,
DO = 5.
Using the
the simjlarity,
28. @]
E' Draw
Draw the height
height from
A as
asshown
shown below.
below.
from A
AADB turns
out to bea
30‐60‐90 triangle.
triangle.
.0.ADB
turns out
be a 30-60-90
AD _ DO
£
= @
D
E
DBB _ O
OE
A
AD
£3
3
3
5
” 44
5
15
A D =z 44‐ =
z 33B
..75
AD
'----
D
3
B
The area
area is
is ~%(6)(3\/3)
The
(6)(3v'3)
--
-----""'- C
6
9\/3
= 9v'3
328
PANDA
COLLEGE PANDA
THE COLLEGE
THE
31.
[g
34.
A
[g Notice
that both outer triangle RQT and
T.
share L
and share
angles and
right angles
have right
QST have
triangle QST
triangle
4 T.
measures,
angle
same
the
have
they
Because
Because they have the same angle measures,
they
similar.
are similar.
they are
B
I 4
R
F
12
D
Q
Q
cC
T
T
correspond
QST correspond
triangle QST
QT in triangle
and W
ST and
Since 5‐7“
Since
RQT,
triangle RQT,
outer triangle
RT in outer
and W
QT and
with CT
with
following
the following
equate the
can equate
we can
respectively,
respectively, we
ratios:
ratios:
sides
similar. The
are similar.
FBE are
and FBE
ADE and
Triangles ADE
Triangles
The sides
the
than the
longer than
times longer
are 3 times
ADE are
triangle ADE
of triangle
Because
FBE
triangle F
sides of triangle
respective sides
respective
BE.. Because
the
triangle,
45-45-90
a
is
triangle ABO
triangle
ABD a 45‐45‐90 triangle, the
then
BE =
let BE
we let
BO is 12\1'2.
length of BD
length
l z fi . ifIf we
= x, then
2ST _ QTg
DE
3x.
DE =: 3x.
RT
QT -‘ RT
QT
x+3x = 12v'2
x+3x:12f2
4x=12\/f_2
4x = 12v'2
xX ==33V2
\/§
ST _ 15
$1
E
15
15 _ 17
17
15
15
z‐
(15) z~ 13.2
ST =
ST
17(15)
13.2
17
Since
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, or 40 ISince
35. I30,31,
I I
32. 7.2 Triangle ABC is a 5-12-13 triangle
triangle
Triangle ABC
= 5). Triangle
(BC =
(BC
ABC is similar
similar to triangle
equal). Using
are equal).
angles are
(the angles
AED
Using this
this
AED (the
triangles
the ttwo
because the
tricky because
similarity
w o triangles
similarity is tricky
following is
The following
orientations . The
different orientations.
have
have different
setup:
correct setup:
a correct
example of a
one
one example
and
L'..BAC and
L'..BDE ~
AC, ABDE
to E,
parallel to
DE
’‐‘_~’ ABAC
DE is parallel
and
6 B0£ and
Therefore , ABDE
L'..BCA . Therefore,
[L'..
BBED
E D~
E ABCA.
are
BAC are
sides of 6ABAC
The sides
similar. The
are similar.
6 BBAC
A
A C are
6+4
means
which means
6 8D£, which
sides of ABDE,
the sides
~ the
:4 = 149
as
~O as
be ‐14‐0
also be
the
must also
p, must
6 BAC, p,
perimeter of ABAC,
the perimeter
AB
AE
A_E_A_B
D
C
DEE ‐ BBC
Thus,
long
6 80£. Thus,
perimeter of ABDE.
the perimeter
as the
long as
12
AE
A_E_e
33.
15
15
33
‐ 55
AE
AE
36
= 36
‐5 == 7.2
z2
5
10
10
4( 12) ~ p ~ 4(16)
(16)
30 ~ p ~ 40
where
integer.
an integer.
where p is an
[fil Since AC
and OF cut through three
into
divided into
are divided
OF are
and D?
AC and
lines , TC
parallel lines,
parallel
the
up the
set up
then set
proportional
can then
We can
parts. We
proportional parts.
multiply .
following
cross multiply.
and cross
equation and
following equation
xX
4
X
xX
= 2v'2
2
x2
x2 = 8
329
CHAPTER
CHAPTER 30 ANSWERS
ANSWERS TO THE EXERCISES
EXERCISES
Chapter23: Circles
Chapter
Circles
CHAPTER EXERCISE:
CHAPTER
EXERCISE:
1.
l. [[)
E The
The circumference
circumference of the
the circle is 2m.
2 m . The
The
square
square divides
the circle
circle into
into four
equal arcs.
arcs.
divides the
four equal
27rr
7rr
. 2rrr
rrr
Therefore,
Therefore, the
the length
length of arc AP
APD
is T = 7
D 1s
6. I6O IBecause
A G is formed
Because [LBBAC
formed from
from the
the
endpoints
diameter,
its
measure
endpoints of a
a diameter, its measure is 90°.
90° .
Since
n d AC = 2, AABC
Since AB = l aand
.6.ABC is a
30‐60‐90
ABAC
= 60°
60°..
30-60-90 triangle
triangle and
and L
BAC =
2. El
[Q]Finding
the radius
radius of each
each of the
2.
Finding the
the small
small
circles,
circles,
7.‑
7. [I]
4
2
2
nr2=367r
rrr
= 36rr
7rr2:97r
rrr 2 = 9rr
r =: 6
rr =
=3
The circumference
The
circle is
circumference of the
the circle
2m = 27r(6)
2m
127r. Because
the equilateral
2rr (6) ~_‐
= 12rr.
Because the
equilateral
triangle splits
splits the
the circumference
circumference of the
triangle
the circle
The
radius of the
The radius
the outer
outer circle is equivalent
equivalent to
to
three radii
radii of the
three
the smaller
smaller circles,
circles, 3 x 3 =
: 9.
The
then rr(9)
The area
area is then
7t(9)22 =
: 8lrr.
817T.
3.
into 3 equal
equal pieces,
into
pieces, arc
arc AB
fl is one-third
one-third of the
the
[I] First,
the radius.
radius.
First, find
find the
circumference: ~
%x
127r =
: 4m
circumference:
x 12rr
4rr .
m22 = 36rr
3671
m
8. ~ The
the circle
The area
area of the
circle is
mr r =
7
= rr(6
7'r(6)2
= 36rr.
3671. The
The shaded
shaded sector
)2 =
sector is
= 6
rr =
lOrr
. circle,
. 1e, which
1 0n:
” =
‐ i S of
entire
o f the
t h e en
tire
cue
w h 1.ch means
means
The
the circle
The circumference
circumference of the
circle is
2rrr
(6) =
2m =
: 2rr
27r(6)
= 12rr.
127r. The
The perimeter
one
perimeter of one
region
radii and
and one-eighth
region is made
made up of ttwo
w o radii
one-eighth
of the
the circumference.
circumference.
if?
36 _ 18
18
5
central angle
angle AC
central
u s t be
ACBB m
must
be % of 360.
18
5 x 360°
_
1‐8 X
360o =
‐ 100°
100o
~
1
6+6+;(12n)
=12+1.57t
6 + 6 + (12rr ) =
12 + l.Srr
8
18
Converting
to radians,
radians,
Converting this
this to
4.@
4.@]
0
lOO
7rr2:497r
rrr2
= 49rr
2 = 49
rr2=49
r=
: 7
rr
180
X
Srr
=9
Wecould’ve
We could've gotten
gotten this
this answer
answer directly
directly by
sticking
The area
area of aa sector
sector is
sticking to radians.
radians. The
1
r 2 8 when
when 8,
the measure
érze
9, the
measure of the
central
the central
2
The
form of a
The standard
standard form
a circle with
with center
center
2
(h,k)) and
and radius
+ (y -‐ k)2
= rr2.
(h,k
radius r is (x -‐ h)2
h) 2 +
k) 2 =
.
So the
equation of the
So
the equation
the circle
circle is
+y2
(x + 2)2
2)2 +
y 2 = 49.
5.
angle,
radians..
angle, is in radians
1
=107r
-$39
r 8=
lOn:
2
2
The arc measure
XE is twice
the
[g The
measure of AB
twice the
1
2
1(6)26
‐_1 lOrr
07r
-(
6)28 =
measure
the inscribed
inscribed angle.
measure of the
Therefore,
angle . Therefore,
60°
-A
60°
11
: 60,wh1chis
° which
' is
' ‐360°
‐
f t he
AB
AB =
60°,
= ‐60
3600
6 of the
circumference.
circumference.
2
189
1071
188 =
= lOrr
5
80=: gn
- 1[
9
330
330
THE
THE COLLEGE
COLLEGE PANDA
PANDA
9. I4, 5, 6, or 71The arc length can be
determined by rOwhen
r0 when 0,
determine
measure of the
the
0, the
the measure
central angle,
angle, is expressed
expressed in radians.
central
radians.
Therefore, the
the arc length
length must
Therefore,
m u s t be
be greater
greater than
than
5
12. E]
@]Circle
Circle P and
and circle U
LI each
each have
have an
an area
area of
7r(3)22 =
= 9rr.
971. To get
get the
shaded region,
we
rr(3)
the shaded
region , we
need
subtract out
o u t the
need to subtract
the unshaded
unshaded portions
portions of
both
APH
U is equilateral,
both circles. Because
Because b.
PHU
equilateral,
AH
PU and
UP are
L
HPU
and AH
L HUP
are both
both 60°, which
which means
means
the unshaded
unshaded sectors
are each
the
sectors are
each one-sixth
one-sixth of
their respective
their
respective circles (60° is one‐sixth
one-sixth of
360°).
(g) ~z 3.92 and
and less
less than
(i)
than 5 ((g)
i) ~z 7.85.
We
could've done
Wecould’ve
done this
this question
question by converting
converting
radians
but the
radians back
back to degrees
degrees but
process
the process
would've taken
taken aa lot longer.
longer .
would’ve
9rr +
9rr ‐97r
+971
10. E]
@]Draw
square connecting
connecting the
Draw aa square
the centers
centers of
each
each circle:
13.
1
1
6
6
(9rr) - (9rr) =
g(97T)‐%(97t)
=157r
15rr
[!] Let y be the angle at the top of the triangle .
2
_y rrr 22:= 247f
rrr
7rr -_ _Jf._
360m
247r
360
rr(6)
rr(6) 2 =
24rr
7r(6)22 -‐ L
firms?
= 2471
360
l = 24
36
36 -_ y_
10
24
10
get the
the shaded
need to subtract
subtract
To get
shaded region,
region, we
we need
out the
the four
four quarter-circles
quarter -circles from
from the
the square
out
square..
The square
square has
has an
an area
area of 8 x 8 =
The
= 64. The
The
four
quarter-circles make
four quarter-circles
make up
up one
one circle with
with
an
area of rr(4)
an area
7r(4)22 =
= 16rr.
167i. The
The area
area of the
the
shaded
shaded region
region is then
then 64
64 -‐ 167f.
l67r.
11.
_ y_
1
12 =
10
12 _ 10
: y
120 =
If
then x and
have to add
add up to 60.
If y is 120, then
and x have
Therefore, x =
Therefore,
= 30.
[£] Unraveling
Unraveling the
the cylinder
cylinder gives
rectangle
gives aa rectangle
14. ~ From
the information
information given,
given, AB =
From the
: 8,
= 4, and
and because
because AC is tangent
BC =
tangent to circle B,
L ACB is aa right
Using the
ZACB
right angle.
angle. Using
the
pythagorean
pythagorean theorem
theorem to find A
C,
AC,
with
base equal
the circumference
and a
a
with aa base
equal to the
circumference and
height equal
equal to
the height
height of the
height
to the
the cylinder:
cylinder:
AC2 +
42 =
+42
= g2
82
h
AC2=48
AC 2 = 48
A c :=44/3
\/§
AC
27rr
2rrr
The
area of the
the cylinder
the
The surface
surface area
cylinder is
is equal
equal to the
area
this rectangle
rectangle plus
plus the
the areas
areas of the
the two
area of this
two
circles at
end .
at either
either end.
The area
area of
b.A
ABC
!( AC )( BC) =
)( 4) =
A
B C = %(AC)(BC)
= !l (4v'3
(4\/§)(4)
= 88\/§
v'3
2
2
2mh
2m 2 = 27r(4)(5)
2rr(4)(5) + 2rr(4)
27rrh + 27'tr2
27r(4)22
15.
[!]The
center ( -‐ 22,, -‐ 44)) and
and radius
radius
The circle has
has center
2. If
If you
you draw
'll see that
that
draw this
this circle oout,
u t , you
you’ll
it's tangent
tangent only
it’s
y‐axis.
only to the
the y-axis.
=
= 4071+
40 7f + 327r
327f
= 727f
7271
=
331
331
CHAPTER 30 ANSWERS
CHAPTER
ANSWERS TO THE
THE EXERCISES
EXERCISES
Chapter 24:
24: Trigonometry
Trigonometry
CHAPTER
CHAPTER EXERCISE:
EXERCISE:
1.
1.
0
Since
(90 -‐ xx),
),
Since cos
cos x = sin
sin(90
cos40
cos 40°° = sin50
sin 50°° = a.
2. []] Since
Since tan
tan x = 0.75 =
z ~,
2, we can
can draw
draw aa
‘
5.
After drawing
drawing the
triangle, we
we let
5. [f] After
the right
right triangle,
let
and the
adjacent side
the opposite
opposite side
the
side be
be m
m and
the adjacent
side
be 1.
be
1.
right
right triangle
triangle such
such that
that the
the opposite
opposite side
side is
is 33
and
and the
the adjacent
adjacent side
side is 4.
4.
m
m
I
1
3
Using the
pythago;ean
orean theorem,
theorem, the
Using
the pyth✓
the
hypotenuse is \/ m
+ 1.
hypotenuse
m2+
1. Therefore,
Therefore,
,.
m
l
4
.
Using
Using the
the pythagorean
pythagorean theorem,
theorem, the
the
hypotenuse
is
5
(this
is
a
3-4-5
triangle)
hypotenuse is 5 (this is a 3 4 ‐ 5 triangle)..
‐ .~=
vm 2 + 1
Vm2+ 1
6.
Therefore, cos
cos xx =
2 4f =
= 0.8
0.8
Therefore,
[I!] The
The fact that
that AB
AB = 55 is irrelevant
irrelevant since
since
the ratios
ratios of the
the same
the
the sides
sides will
will always be
be the
same
proportional triangles.
for proportional
triangles. Instead
Instead of actually
actually
trying
out the
thellengths
the sides,
Sides,
trying to
to figure
figure out
lengths of
of the
let 5 use
use a
ea51er to
to work
With..
let's
a triangle
triangle that
that'ss easier
work with
5
3, [QJ
El Since
Since sin 9 2 cos (90 _ 9) and
sin 0 = cos (90 - 0) and
3.
ss1nx=
m x
_ .
cost) =
‐ sin
sm(90
9),
cos0
(90 -‐ 0),
B
sine0 + cos(90
cosG
+sin(90
9) =
sin
cos (90 -‐ 0) + cos
0+
sin (90 -‐ 0)
sin0+sin9+c056+cose
sin
0 + sin 0 + cos 0 + cos 0 =
225in9+2cos€
sin 0 + 2 cos 0
3
3
4. I25 IDrawing
Drawing the
triangle,
4.
the triangle,
A
A
4
4
C
Using the
pythagorean theorem,
theorem, BC
BC =
: 5 (it's
(it’s
Using
the pythagorean
a
3‐4‐5 triangle).
a 3-4-5
triangle) .
30
4
..mBB + c o sBB ‐ 4§+§
33 ‐ 5
77 ‐ 1 . 4
sSm
+ COS = S + S
=S
= 1.4
C
C
B
5
cosA =
= 5
‐
cosA
6
6
7.
7‑ [IT]
l
AC z §5
A_C
30
6
.
1
= -_
S 'l n x =
4
smx
4
i3
_ 11
-4
BC
BC
4
AC =
= 25
25
Cross
get BC
BC:= 12.
Cross multiply
multiply to
to get
332
THE COLLEGE
PANDA
COLLEGE PANDA
AC = ‐3 .
r:;-i ..
AC
0
7 SmcecosxO=
‐3
6,
Since AC z= 6,
=-.4.SinceAC
- , ‐= 4'AB
13. L.:JSmcecosx
5
12
, we can let
8. , 1 / Since tan B = 2.4 =
3
5
AC
12and
pythagorean
the pythagorean
Using the
AB = 5. Using
and AB
= 12
AC =
theorem,
w o triangles
are
triangles are
the ttwo
= 13. Since the
BC 2
theorem, BC
similar,
similar,
AB
AB
5
BC
13
4 AB
3
Therefore,
= 7.
Therefore, k =
are
DBE are
and DBE
ABC and
triangles ABC
that triangles
Notice that
14. j 12.s INotice
equivalent
is
BAC
angle BAC equivalent
means angle
similar,
which means
similar, which
tangent of L'.
the tangent
to angle
[ BBAC
A C is
BOE . Since the
angle BDE.
also 1.25.
L'.BDE
1.25, the
B D E is also
tangent of é
the tangent
Opp
op~
adj =: 1.25
adJ
sin (5m -‐ 12)
sin 58 =
: sin(5m
5
= 5Sm
m -‐ 112
2
588 =
BE
BE
BE
1.25
= 1.25
DE '‐
7
m
700 =5Sm
m=
1
= 144
(1.25)( 10) = 12.5
(1.25)( DE) := (1.25)(1o)
BE
BE = (1.25)(DE)
@]From the coordinates,
AB = 5 - ( - 3) = 8
the
Using the
12 - ((-‐ 33)) =
BC := 12‐
and BC
and
: 15. Using
AC ,
theorem to find AC,
pythagorean theorem
pythagorean
it's
because it’s
measures 90° because
B C measures
@][L'.AABC
15. [E
triangle
semicircle. Therefore,
inscribed in aa semicircle.
inscribed
Therefore, triangle
be AB
height be
right triangle.
a right
ABC is a
ABC
triangle. Let the height
AB
hypotenuse
the
Since
BC.
be
base
the
and
and the base be
hypotenuse
1,
AC
AC = 1,
AC2
+ 3c2
BC2
AB 2 +
= A32
AC 2 =
AC2=82+152
AC 2 = 82 + 152
AB
sin0 == AB
sine
Ac2
: 2289
89
AC 2 =
=
ccos0
ost) = B
C
BC
A C =: 117
7
AC
BC _ 15
.
BC
Finally,cosC
AC ‐= 1‐7..
Finally, cos C ‐= E
17
Area of
= ~%(BC)(AB)
( BC )( AB )
triangle =
of triangle
1
.
= 1§(cosl9)(sm
(cos0 )( sin09))
=
8
.
o
. 0o . 0o 8
- x 0 ))== ssin
cos (90° ‐x
m x ,, ssin
mx =
_‐_ fi' .
@]Since
11. @Smcecos(90°
17
be
angle x be
the angle
50
opposite the
side opposite
the side
can let the
we can
So we
the
Using the
8 and
be 17. Using
hypotenuse be
the hypotenuse
and the
adjacent to
pythagorean
side adjacent
theorem, the side
pythagorean theorem,
Finally,
82 = 15. Finally,
J17 2 -‐ 82
has length
angle x has
angle
length \/172
adjacent
15
15
adjacent
:
‐ :
‐ ,
cosx 0o =-~-17
hypotenuse
cosx
hypotenuse
17
0cos 0
_ sin
sinGcosH
_
16.
2
2
the
asking for the
basically asking
question is basically
This question
[g This
0. For
equal cos 6.
can equal
0 can
sin 9
which sin
quadrants in which
quadrants
For
the same
have the
must have
be equal, they must
them to beequal,
them
same
since sine
sign.
sine is
option II since
out option
rules out
That rules
sign. That
cosine
while
quadrant
second
the
in
positive
positive
second quadrant while cosine
are
cosine are
and cosine
sine and
quadrant I, sine
negative. 1n
is negative.
In quadrant
both
sine is equal
equal to cosine
cosine
and sine
positive, and
both positive,
[I) Draw a triangle . Since the sine of one of
J;,
angle s is ?,
acute angles
the acute
the
3
4
pythagorean
theorem,
pythagorean theorem,
./28= 2/7.
,/~64--- 3-6 := \/2_8=2\/7.
JB 2 - 62 = \/64‐3
BC = \/82-62
BC=
9. [!Ij Since COS X = sin (90 - x),
equation,
an equation,
up an
Setting up
sin 58°. Setting
cos32
cos
32°° = sin
12.
4
AB
= §(6)
8. Using
the
Using the
= 8.
(6) =
(AC ) =
= §(AC)
AB =
‐ =z -13‐
cos B =z -BC
N = :cosB
cos N
10.
4
side
can let the side
we can
hypotenuse
the hypotenuse
and the
opposite
v'3and
be \/3
angle be
the angle
opposite the
side
the side
theorem, the
pythagorean theorem,
the pythagorean
Using the
be 2. Using
be
length
adjacent
has length
angle has
the angle
adjacent to the
45-45-90
your 45‐45‐90
when
(remember your
when 0 := 45° (remember
are
cosine are
and cosine
sine and
lll, sine
triangle?).
quadrant III,
1n quadrant
triangle?). In
cosine
equal to cosine
both
sine is equal
and sine
negative, and
both negative,
quadrant
third quadrant
when
(this is the third
0 = 225° (this
when 6
quadrant) .
equivalent
the first quadrant).
equivalent of 45° in the
a
that this
y'3)2 =
J22
22‐- ((x/3)2
= l.1. Notice
Notice that
this is a
complete ,
triangle complete,
30‐60‐90
triangle. With the triangle
30-60-90 triangle.
angle is then
acute angle
other acute
sine of the other
the sine
the
then
opposite
opposite _ 1
hypotenuse ‐- 2
hypotenuse
2'
333
CHAPTER 30 ANSWERS TO THE EXERCISES
Chapter 25: Reading Data
CHAPTER EXERCISE:
1.
II] We estimate
the total commute time for each point:
I
Point
A
B
C
D
Commute Time
25 + 60 = 85
38 + 40 = 78
45 + 80 = 125
80 + 20 = 100
I
“‑
‑
Even
though the
represents the
the greatest
time .
Even though
the times
times were
were estimated,
estimated, it's
it’s clear
clear that
that C represents
greatest commute
commute time.
II] The vertical distance between the points at 2004 and 2006 is the smallest among the answer choices.
3. II] The points corresponding to July through September are the highest in both 2013 and 2014.
2.
150 - 33 150
0
250 - 5 - 60 1/o
lower than
(to the
5. [I]
E San
San Diego
Diego is the
the only
only city for which
which the
the estimated
estimated bar
bar is lower
than (to
the left
left of) the
the actual
actual bar.
bar.
6. [I]
1 Both
Both line
line graphs
graphs go
go downward
downward every
every year.
year.
E The lowest
lowest point
point with
with respect
respect to the
the y-axis
y-axis is at
at a
under 40
40 years
age.
7. @The
a little
little under
years of age.
A The
The graph's
graph’s minimum,
minimum, 16, must
must be
be the
the weight
weight of the
when empty.
empty. The
graph’s maximum,
maximum,
8. [}]
the truck
truck when
The graph's
must be
be the
the weight
weight of the
the truck
truck at
at maximum
maximum capacity.
capacity. Subtract
Subtract the
the two
t w o to get
get the
the truck’s
maximum
30, must
truck's maximum
capacity, 30
30 -‐ 16
16 = 14.
capacity,
9.
II] From 2010 to 2011, the percent
~ -~
¥
40
decrease was
1
‐}1
4 =z
: =
.
(percent decreases
are negative)
negative)
-‐25%
25% (percent
decreases are
From
From 2013 to 2014, the
the percent
percent increase
increase was
was
2 5-‐ 220
0 = !1 = 25ox
0
25
20
_Z_25/o
20
4
°
ag _z
l27 120
2
lO. li_J
180
=
3
3 180 ‐ 3
11.
IConsole
A generated
generated 250,000
250,000 x 100 =
:
[[j
Console A
12.
in Quarter
Quarter 3,
3, Company
Company Y's profit
profit was
was about
about 66 million
million and
Company X's
X’s profit
was about
12million
million
[I) In
and Company
profit was
about 12
$25, 000, 000. Console
2 5 , 000 x 150 =
: $33,750,000.
$33, 750,000.
$25,000,000.
Console Bgenerated
B generated 225,000
Console D generated
generated 125,000
125,000 x 250 =
: $31,250,000.
$31,250,000. Console
50,000 x 300 = $15,000,000.
$15,000,000.
Console E generated
generated 50,000
Console
Console B generated
generated the
the most
m o s t revenue.
revenue.
Console
(twice Company
Company Y's).
Y’s). In no
no other
other quarter
quarter was
w a s Company
Company X's
X’s profit
profit as
as close
being twice
twice Company
(twice
close to being
Company Y’s.
Y's.
334
THE COLLEGE
PANDA
THE
COLLEGE PANDA
13.
13.
15+
=
[fJ Alabama
Alabama spent
spent a
a combined
combined 15
+ 2.5
2.5 =
17.5 billion.
billion. Alaska
7.5 +
+ 7.5
7.5 =
= 15
15billion.
Arizona spent
spent
17.5
Alaska spent
spent 7.5
billion. Arizona
12.5 +
+ 7.5 =
= 20billion.
Arkansas
spent
10+
5
=
15billion.
Arizona
spent
the
most.
20 billion. Arkansas spent 10 + 5 = 15 billion. Arizona spent the most.
14. I44 / During
first two
Jeremy answered
answered 44 calls
total of
of 22 xx 4
During the first
two hours,
hours, Jeremy
calls per
per hour
hour for
for aa total
4 = 88 calls.
calls. During
During
the
hours, Jeremy
Jeremy answered
answered 8 calls per
per hour
During the
the next
next three
three hours,
hour for a
a total
total of 3 x 8 = 24 calls. During
the final
two
hours,
Jeremy
answered
a total
total of 2 x 6 =
t w o hours, Jeremy answered 6 calls
calls per
per hour
hour for a
z 12
12 calls. He answered
total of
answered a
a total
+ 24
24 +
+12
= 44
44 calls.
8+
12 =
that it takes
takes Greg's
glucose level
level 2.5 hours
hours to return
return to its
its initial
initial value
value (140
15. [}] From
From the graph,
graph, we
we can
can see that
Greg’s glucose
mg/
dL)
after
breakfast
and
8
4
=
4
hours
to
return
to
its
initial
value
(also
140
mg/
dL)
after
mg/dL) after breakfast and ‐ = hours return
initial value
mg/dL) after lunch
lunch..
. 5=: 1.5
4 -~ 22.5
16. ~
At 30
per hour,
hour, Car
Car X
X gets
gets 25
for 5
5 hours
hours at
at 30
per hour
hour covers
covers a
a
16.
E At
30 miles
miles per
25 miles
miles per
per gallon
gallon.. Driving
Driving for
30 miles
miles per
total
distance
of
5
x
30
= 150 miles
total distance
m i l e s .,
150 miles
miles x
11gallon
gallon
.
= 6 gallons
25
miles : 6 gallons
25 miles
335
-
ANSWERSTO THEEXERCISES
‘
\‘ t.
f½PTER30
HAPTER 30 ANSWERS TO THE EXERCISES
2.6:Ptobability
‘/ 'Chaptet
thapter 26:
Probability
CHAPTEREXERCISE:
EXERCISE:
CHAPTER
1.‑
Stop sign violations committed by truck drivers = 39 ~ 0.433
M
= 3.9. x 0.433
Stop sign
90
Stop
sign violations
violations
90
+
2.
The percentage
of silver
is 100
= 23. Red
and silver
silver make
make up 20 + 23 =
= 43
2. @]
ElThe
percentage of
silver cars
cars is
100 -‐ 20
20 -~ 33
33 ‐- 10 ‐- 14 =
Red and
percent of the
the cars.
cars.
percent
3.@J
3.
El
Plumbers
least 4 years
Plumbers with
with at
at least
years of experience
_
+ ‐45,376
45,‐376‐ z
experience =
40,083
~ 0.46
0.4
_ 40,
‐ ‐083‐ +
6
All plumbers
183,885
plumbers
183,885
4.‑
4.
@
083 + 45,
376 z
= 40,
40,083
45,376
~ 0.22
_
______ =
0 22
182,
410 +
+ 208,
757
182,410
208,757
Plumbers with
with at
at least
least 4 years
years of experience
experience
Plumbers
Workers with
with at least
years of experience
Workers
least 4 years
experience
.
_
_
‐
‐
_
_
_
.
5. Cg
5.‑
Games
Games won
w o n as
asunderdogs
10 _ 22
underdogs _ 10
Games
played as
Games played
asunderdogs
45 _ 9
underdogs
45
6. CgFilling
Filling in the
the table,
6.
table,
W e e k l]
Week
Week2 \ Week3
Week2
Week3
Weel<4
Week4
Total
Total
20
55
150
springs
Box springs
35
35
Mattresses
Mattresses
47
61
68
22
198
Total
82
101
88
77
348
40
Box spring units sold during weeks 2 and 3 _ 40 + 20 _ 2
All box
box spring
spring units
units sold
-_ 150 - 5
!~
. 15W
. 29 x 0.28. For Russia, the probability is g z 0.39. For Great Britain,
7.
7. @ For
For the
the USA,
USA, the
the probability
probability is
: ~ 0.28. For Russia, the probability is
~ 0.39. For Great Britain,
!:
1
. . . 19
the probability
probability is
15@ ~
z 0.29.
0.29. For
For Germany
Germany,, the
the probability
the
probability is
is
0.32. The
~ 2~ 0.32.
The country
country with
with the
the highest
highest
[11‐2
probability
probability is
is Russia.
Russia.
8.
8.
[I]
Cartilaginous fish
fish species
in the
Philippines Cartilaginous
fish species
species in
Cartilaginous
species in
the Philippines
Cartilaginous fish
in New
New Caledonia
Caledonia
Total fish
species in
Total
Total
fish species
in the
the Philippines
PhjJippines
Total fish
fish species
species in
in New
New Caledonia
Caledonia
:
336
336
400
_
300
_ 1
1 1
1_
300
-,--,----------400+8OO
5 _55 _
400 + 800 3oo+1,2oo
300 + 1,200 ‘ 3
2
2
E
15
‐‑
PANDA
COLLEGE PANDA
THE COLLEGE
THE
table,
the table,
Filling in the
9. [g Filling
9.
_
Lightning-caused
Lightning-caused fires
Human-caused fires
Human-caused
Total
East Africa
55
ss
65
65
120
South Africa
30
70
100
Total
85
135
220
‐_
Africa _ 65 _ 13
East Africa
-caused fires in East
Human
13
Human-caused
Africa
24
East Africa
Fires
Fires in East
_ 120 _ 24
10. []]
Line A
Assembly Line
from Assembly
Defective from
Defective
Defective
Defective
11.
300
_@
‘ 8800
00‘
-E
3
88
[Q]
less
17
family members
with 2 family
Duplex with
12
+ 12_
Duplex
members or less_
22 +
17
=
23
46
Duplex
Duplex
46
23
(450 x 0.08) +
Chemical A is (450
contaminated with
samples contaminated
number of samples
total number
The total
12. []] The
with Chemical
+ (550 x 0.06) =
= 69.
Contaminated
69 = _
= ~
samples _
Contaminated samples
0 069
= 0.069
1, 000
samples
All samples
All
_ 1,
13.
and
virus and
the virus
have the
don't have
who don’t
patients who
indicators for patients
positive indicators
gives positive
when it gives
incorrect when
test is incorrect
The test
[fil The
occurrences.
= 80 occurrences.
total of 30 + 50 =
do, a total
patients who
indicators for patients
negative indicators
negative
who do,
80
8
‐ = =~=
‐ :
~
100
1000
%
8%
8
weren't
patients who
number of patients
The number
90 + 33 = 30. The
pill is 90+
sugar pill
the sugar
by the
cured by
patients cured
number of patients
The number
14. []] The
who weren’t
30 x ;g =
pill is 30
sugar pill
the sugar
cured
z 75.
cured by the
Drug
Cured
Cured
90
cured
Not cured
Not
30
75
Sugar
Pill
Sugar Pill
25
Given aa sugar
sugar pill
pill and
30
_2
=2
30
cured _
and cured_
Given
7
75
- 30 +
pill
Given aa sugar
Given
sugar pill
=30
+ 75
I I
be x.
equipment be
gym equipment
prefer gym
who prefer
seniors who
number of seniors
the number
Let the
15. 240 Let
L
x
X
X
_
1
1
+
+ 160 _ 3
Cross
multiplying,
Cross multiplying,
= xX ++ 160
3x =
2
60
=1160
2xx =
x :=880
0
X
school.
the school.
seniors at the
+ 160 := 240 seniors
are 80 +
There are
There
337
CHAPTER 30 ANSWERS TO THE EXERCISES
Chapter 27: Statistics I
CHAPTER EXERCISE:
1.
is
class is
second class
the second
heights in the
sum of the heights
The sum
= 882. The
63 =
the first class
heights in the
[f] The sum
sum of the heights
class is 14
14 x 63
2.
[f]
average
The average
2,310. The
then 882 +
class is then
combined class
21 x 68 = 1,428.
1,428. The ssum
u m of the
the heights
heights in the
the combined
+ 1428 = 2,310.
height
height is
Sum
310 =
2,310
= 2,
heights
the heights
Sum of the
66
+
14
number of students
students I 14+ 21 = 66
Total number
The sum of all five of Kristie's test scores is 5 x 94 = 470. The sum of her last three test scores is
=
test scores:
her first ttwo
the sum
between these
difference between
The difference
3 x 92 = 276. The
these ttwo
w o sums
sums is the
s u m of her
w o test
scores: 470 -‐ 276 =
194
.
.
= 97.
then ‐2- =
1s then
scores is
test scores
her first ttwo
w o test
average of her
194. The average
2
3.
I]] A range of 3 days means
4.
I]] Because there are 20 editors,
5.
[QJThe median
6.
[fil
the difference between the longest shelf life and the shortest shelf life among
the
about the
nothing about
range says
The range
days. The
28 days.
vs . 28
days vs.
be 10
could be
3. This could
units is 3.
the
10days
days vs.
vs. 13
13days
days or
or 25
25days
says nothing
the units
median.
mean
mean or median.
the median is the average of the 10th and 11th editors' number of books
which
year, which
last year,
books last
15 books
and 15
between 11 and
read between
both read
the 10th
graph, the
the graph,
From the
read. From
10th and
and 11th
11th editors
editors both
read.
is 12.
and 15
between 11 and
choice between
answer choice
only answer
be between
must also
means
the average
average must
also be
between 11 and
and 15. The only
15is
means the
score, designated by the line segment in the middle of the "box", is approximately 83.
plot, is approximately
the plot,
end of the
the left end
at the
segment at
designated by the line
score, designated
individual score,
lowest individual
The lowest
The
line segment
approximately
them is 83
between them
83 -‐ 72 =
= 11.
difference between
72. The difference
(18x6)+(19x3)+(20><5)+(21x4)+(22x2)+(23><3)+(24x1)
(18 X 6) + (19 X 3) + (20 X 5) + (21 X 4) + (22 X 2) + (23 X 3) + (24 X 1) :
24
‐ 20.25
20.25
24 =
= 24
24
7. I67 IThe median is represented by the average of the 14th and 15th days, both of which are 67°F.
8.
[QJThe standard
deviation decreases the most when the outliers, the data points furthest away from the
the Vosges.
and the
Rhone and
the Rhone
are the
here are
outliers here
The outliers
mean,
removed . The
are removed.
mean, are
9.
10.
[QJBy definition,
at least half the values are greater than or equal to the median and at least half the
the median.
values
are
less
than
median.
equal to the
than or equal
values are
out for Bus
spread out
are more
themselves are
times themselves
same, the
the travel
travel times
m o r e spread
the same,
are the
frequencies are
though the frequencies
Even though
GI]Even
times
deviation of travel
standard deviation
the standard
Therefore, the
together . Therefore,
closer together.
are much
B. The travel
much closer
travel times
times for Bus A are
travel times
for Bus A is smaller.
smaller .
11.
Company
and Company
Company A and
both Company
pounds for both
the 10th
weight is represented
represented by the
10th kayak
kayak (47 pounds
median weight
[f] The median
both companies.
same for both
B). The
companies.
the same
weight is the
median weight
The median
12.
order,
Arranging the
scores in order,
the scores
I]] Arranging
75, 83, 87, 87, 90, 91, 98
75,83,87,87,90,91,98
+ 9900 + 9911 +98
median is
The median
87.
mode iis 8
s‐7755‐++ 883
‐3++78877‐++ ‐88:7‐+
‐ +‐ ‐ +‐ 98‐ ~
at: 87.3. The mode
s7. The
i aalso
slso 87.
true .
and IIll
lI is false, and
75 =
I I is true.
numbers, I is false, I]
these numbers,
From these
= 23. From
The
average iis
The average
range
range is 98 -‐
338
The
The
THE COLLEGE
PANDA
THE
COLLEGE PANDA
13.
I]]
(5x2)+(6x1)+(8x4)‐+‐(9><2)+(10><1)_§_
Mean = (5
x 2) + (6 x 1) + (8 x 4) + (9 x 2) + (10 x 1) = 76 = 7.6
Mean :
7.6
2 ++ 1 ++ 44 ++ 2 ++ 1
“ 1100 ‘
n g e= l10e- ‐5 =s 5t
Range
14.
[Q]
[E] Before the 900-calorie
900‐calorie meal
added, the
the median
median is the
the 5th
meal is added,
the average
average of the
5th and
and 6th
6th meals
meals (550),
(550), the
the
mode
mode is 550, and
and the
range is 900 -‐ 500 =
z 400. After
becomes
the range
After the
the 900-calorie
900-calorie meal
meal is
is added,
added, the
the median
median becomes
the 6th
the
6th meal
meal (still
still 550, and
of them
them change.
change.
(still 550), the
the mode
mode is still
and the
the range
range is
is still
still 400. None
None of
15. [Q]
The median
E] The
represented by the
10th class
B is
median in School
School A is represented
the 10th
class (4
(4 films) and
and the
the median
median in
in School
School B
is
represented
represented by the
8th class
The median
same in
schools.. N
o w we
the 8th
class (also
(also 44 films). The
median is
is the
the same
in both
both schools
Now
we calculate
calculate the
the
means:
means:
1 x 2 + 2 x 3 + 3 xx 44 ++ 44 xx 5
5~
Mean in School A =
5 ++ 55 xx s
MeaninSchoolA
: 1 x 2 + 2 x 3 + 3 19
z3.42
3 .42
19
MeaninSchoolB=1X1+2X2+31x53+4><4+5x5¢=1367
M
. Sch oo l B = 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 + 5 x 5 ~ 67
ean m
3.
15
The mean
mean is greater
The
B.lntuitively,
distribution for
for School
School A
A has
has aa
greater in School
School B.
Intuitively, this
this makes
makes sense
sense because
because the
the distribution
higher proportion
proportion of the
the smaller
higher
numbers 1,
1,2,
asshown
These smaller
numbers pull
smaller numbers
2, and
and 3 as
shown by the
the chart.
chart. These
smaller numbers
pull
down the
the mean.
mean .
down
16.
I]] Before
removed, the
Before the car
car is removed,
the median
median is represented
represented by the
is removed,
the 8th
8th car
car (23 mpg).
mpg). After
After the
the car
car is
removed,
the median
median is represented
the
represented by the
average of the
the 7th
stays the
the
the average
7th and
and 8th
8th cars
cars (still
(still 23
23 mpg).
mpg). So
So the
the median
median stays
same. However,
However, the
same.
standard deviation
We’re removing
removing aa data
data point
higher
the mean
mean and
and the
the standard
deviation both
both decrease.
decrease. We're
point higher
than
e a n decreases.
We’re also
the spread
spread in
in the
the data
so the
than all
all the
the others
others so
so the
the m
mean
decreases. We're
also reducing
reducing the
data so
the standard
standard
deviation
deviation decreases.
decreases.
17.
[J] First,
First, it's
it’s easy
easy to see
see that
that the
mean will
will decrease
decrease since
the mean
since we're
we're replacing
replacing the
the maximum
maximum data
data point
point with
with
a
N o w before
= 45. After
range
a minimum.
minimum. Now
before the
the replacement,
replacement, the
the range
range is
is 90
90 -‐ 45
45 =
After the
the replacement,
replacement, the
the range
65 ‐- 20
20 = 45, so
is 65
so the
range remains
remains the
same.
Before
the
replacement,
the
median
is
represented
the range
the same.
replacement, the median is represented by
by
the average
average of the
the
and 10th
10th cars
cars (57). After
After the
replacement, the
median is
is represented
represented by
10th car
the 9th
9th and
the replacement,
the median
by 10th
car
(still 57, don't
don’t forget
forget to count
the replacement
replacement as
value). The
The median
(still
count the
as the
the first value).
median also
also remains
remains the
the same.
same.
Therefore,
the mean
Therefore, the
most.
mean changes
changes the
the most.
18. ~ To construct
construct the
the box
box plot,
plot, we need
know the
and maximum
number of lectures
need to know
the minimum
minimum number
number and
maximum number
lectures
given, the
the median,
median, the
given,
quartile, and
and the
the third
third quartile.
table
indicates
that
the
is 12
12
the first quartile,
quartile . The
The table indicates that the minimum
minimum is
and the
and
the maximum
maximum is 40.
The median
The
the 90-;90+ 2 =
: 45th
45th and
we can
count up
up in
in
median is the
the average
average of the
and 46th
46th professors.
professors. From
From the
the table,
table, we
can count
order of the
the number
number of lectures
order
see that
both gave
gave 25
25 lectures,
lectures, so
so
lectures given
given and
and see
that the
the 45th
45th and
and 46th
46th professors
professors both
the median
the
median is 25.
To find
median to split
split the
into ttwo
w o halves:
the first
find the
the quartiles,
quartiles, we
we use
use the
the median
the data
data into
halves: the
first 45
45 professors
professors and
and the
the
last
last 45
Note that
the median
45 professors.
professors . Note
that since
since the
median is
is the
the average
average of
of the
the 45th
45th and
and 46th
46th professors,
professors, it’s
it's already
already
"excluded" from
"excluded”
from either
either half.
half.
The
The first quartile
the first 45
by the
22.5 ➔
‐) 23rd
23rd
quartile is the
the median
median of the
45 professors,
professors, represented
represented by
the 45
45 7+ 2
2 : 22.5
professor. From
From the
professor.
the table,
table, the
the 23rd
23rd professor
professor gave
gave 15
the first quartile
15.
15 lectures,
lectures, so
so the
quartile is
is 15.
45 professors,
The third
third quartile
quartile is the
the median
median of the
the last
last 45
professors, represented
represented by
by the
the 45
45 7+ 2
2 = 22.5
22.5 ‐+
➔ 23rd
23rd
professor
the 45th
or the
45+
23
:
68th
professor
overall.
Again
we
count
up
in
the
table
professor from
from the
45th professor,
professor, or
the 45 + 23 = 68th professor overall . Again we count up in the table
and see
and
is 28.
see that
that the
the 68th
68th professor
professor gave
gave 28
28 lectures,
lectures, so
so the
the third
third quartile
quartile is
Now
N
o w that
(min: 12, ql:
q l : 15, med:
max: 40), we can
see that
the correct
that we have
have these
these statistics
statistics (min:
med: 25, q3:
q3: 28, max:
can see
that the
correct
boxplot is the
boxplot
one in answer
answer choice
choice C.
C.
the one
339
CHAPTER 30 ANSWERS
CHAPTER
ANSWERS TO THE
THE EXERCISES
II
Statistics 1!
Chapter 28: Statistics
EXERCISE:
CHAPTEREXERCISE:
CHAPTER
the x-axis is 19.
along the
value along
best fit when
line of best
above the line
points above
are 2 points
1. [I] There
There are
when the value
2.
information.
irrelevant information.
size of 400 is irrelevant
sample size
that the sample
Note that
[I]Note
be a
let xx be
we 'll let
easier, we’ll
things easier,
make things
To make
percentage later,
a percentage
convert itit to a
and convert
decimal for nnow
decimal
o w and
later,
3, 300x =
: 66
3,300x
x=
z 0.02 = 2°/o
2%
X
3.
x-value is 75.
the x-value
when the
55 when
a y-value
gives a
best fit gives
line of best
Cf]The line
y‐value of 55
4.
that's
since that’s
freshman class since
university's freshman
students from the university’s
conducted with
First, the survey
survey should
should be
be conducted
with students
Cf]First,
results.
the results.
valid the
more valid
sample, the more
target. Secondly,
intended target.
the intended
Secondly, the larger
larger the sample,
5.
Candidate B
votes . Candidate
500,000 =
~~~ x 500,000
expected to receive
Candidate A is expected
proportions, Candidate
Using proportions,
receive %3
: 220,000 votes.
GI]Using
receive 280,000 ‑expected to receive
Candidate B is expected
So Candidate
~:~ x><500,000 = 280,000 votes
receive £8
votes.. So
expected to receive
is expected
votes .
more votes.
220,000
220, 000 := 60, 000 more
6.
time
shopping time
average shopping
case, it's
value of x is 0. In this case,
the value
value of y when
They-intercept
GI]The
y‐intercept is the value
when the
it’s the average
discount).
(no discount).
0% (no
discount is 0°/o
store discount
the store
when the
when
revenue
increase in revenue
the increase
it's the
slope, it’s
positive slope,
a positive
best fit has
the line
Because the
run. Because
over run.
The slope
7. GI]The
slope is rise
rise over
line of best
has a
expenses
advertising
and
revenue
both
because
that
Note
expenses.
for every
dollar
increase
in
advertising
expenses.
Note
that
because
both
revenue
and
advertising
expenses
advertising
increase
dollar
every
have no
and have
out and
the graph,
dollars in the
thousands of dollars
expressed in thousands
are
graph, they
they cancel out
no effect on the interpretation
interpretation
are expressed
(8).
isn't (B).
answer isn’t
the answer
why the
of the slope.
That's why
slope . That's
increase in box office
slope, it's the increase
positive slope,
has a positive
fit has
over run
8. []] The slope
slope is rise over
run.. Because
Because the line of best
best fit
minute increase
sales
sales per
per minute
increase in movie
movie length.
length.
expected number
case, it's
In this case,
value of x is 0. ln
value of y when
y-intercept is the value
[QJThe y‐intercept
9. @
when the
the value
it’s the expected
number of
.
prize)
cash
(no
dollars
O
is
prize 0 dollars (no cash prize).
cash prize
mistakes
when the cash
made when
mistakes made
10.
calories . At
are 340 calories.
there are
grams of fat, there
best fit. At 20 grams
the line
slope of the
line of best
asking for the slope
question is asking
[I]This question
points,
slope from ttwo
Calculating the
25
calories. Calculating
the slope
w o points,
are 380 calories.
there are
grams of fat, there
25 grams
= 8
40 _
= 419
380 -‐ 340 _
8
25 -‐ 20
20 _ 5 _
11.
size,
sample size,
From the sample
population . From
larger population.
sample to the larger
learn from the sample
you learn
what you
Apply what
[I]Apply
.
.
Car
Speeding Violations _ 83
Car Speeding
total of 2,000:
proportion to the
same proportion
apply this same
Total Vi
Violations
‐ @' . N
o w we can
can apply
the state
state total
2,000:
Now
=
.
li
284
ota I 10 1ations
83
282 Xx2,000~585
2,000 ~ 585
284
12.
closest to
point closest
represented by the point
best fit is represented
line of best
predicted by the line
best predicted
yield is best
whose yield
oat field whose
[I]The oat
that field.
applied to that
nitrogen applied
amount of nitrogen
which is the amount
the line.
That point
x-value of 350, which
an x-value
has an
point has
line. That
340
THE COLLEGE
COLLEGE PANDA
PANDA
13. IE
[Q]The point
line of best
best fit is at an
x~value of 7. The
number of
point farthest
farthest from the
the line
an x-value
The total
total number
of seats
seats at
at the
the food
food
court
represented by this point
court represented
80 = 560.
point is 7 x 80
14. [Q]
[B To draw
reliable conclusion
about the
the effectiveness
e w vaccine,
vaccine, the patients
m u s t be
be
draw a
a reliable
conclusion about
effectiveness of the
the n
new
patients must
randomly assigned
randomly
answer (D) leads
Note that
answer
assigned to their
their treatment.
treatment. Only
Only answer
leads to
to random
random assignment.
assignment. Note
that answer
does not
because the patients
(C) does
not because
are allowed
allowed to group
group themselves
patients are
themselves as
as they
they desire.
desire . For example,
example, three
three
friends might
friends
want to remain
the same
same group,
group, leading
n
o
t
random.
might want
remain in the
leading to assignment
assignment that
that is
is not random.
15.
[g
16.
[Q]
The lower
lower the
the standard
lElThe
standard deviation
(variability), the
deviation (variability),
the lower
lower the margin
margin of
of error.
error. Selecting
Selecting students
students who
who
Answer
wrong because
Answer (A) is wrong
it’s possible
possible that
produced in
in Week
Week 11 had
an
because it's
that most
most of the
the basketballs
basketballs produced
had an
air pressure
pressure of over 8.2 psi.
air
know for sure
sure.. Answer
Answer (B) is
wrong because
because
psi. Likewise
Likewise for Week 2. We don’t
don't know
is wrong
it's too
too definite.
it’s
Just because
because the
the sample
sample means
means were
mean the
which
definite. Just
were 0.5 psi
psi apart
apart doesn’t
doesn't mean
the true
true means,
means , which
would
take into
would take
into account
account all the
the basketballs
produced in Week 1 and
and Week 2, were
basketballs produced
were also
also 0.5 psi
psi apart.
apart.
That's
there 's aa margin
That’s why
why there’s
margin of error
the samples
samples.. Answer
suggest
error for the
Answer (D) is
is wrong
wrong because
because the
the samples
samples suggest
the reverse:
reverse: the mean
the
mean air
e a n air
a i r pressure
for Week 2
afr pressure
pressure for Week 1 (8.2 psi)
psi) is greater
greater than
than the
the m
mean
pressure for
psi). Answer (C)
(7.7 psi).
size,
the
lower
the
margin
of
error. The
The
(C) is correct
correct because
because the
the greater
greater the sample
sample size,
lower the margin
sample from Week 11 had
had a
sample
lower margin
margin of error
error than
the
sample
from
Week
2.
a lower
than
sample
2.
are following
following the
the same
same daily
are
plan will
deviation because
they are
are
daily diet
diet plan
will likely lead
lead to the
the lowest
lowest standard
standard deviation
because they
likely to be
servings of vegetables.
The other
other answer
would result
result in
in
be eating
eating the
the same
same number
number of servings
vegetables . The
answer choices
choices would
much more
more variability.
much
variability .
17.
[g Answer
best expresses
Answer (C)
(C) best
expresses the
the meaning
meaning of a
statistical
a confidence
confidence interval,
interval, which
which applies
applies only
only to the
the statistical
mean
m
e a n and
and does
does not
n o t say anything
about blue-spotted
wrong
anything about
blue-spotted salamanders
salamanders themselves.
themselves . Answer
Answer (D) is
is wrong
because the
the study
involved only
because
study involved
blue-spotted
salamanders, nnot
o t all salamanders.
only blue
-spotted salamanders,
salamanders .
18.
I]] The most
most that
that we can conclude
conclude is that
that there
there is a negative
food and
negative association
association between
between the
the price
price of food
and
the population
population density
density in U
S . cities (as one
one goes
goes up,
WeCANNOT
U.S.
up, the
the other
other goes
goes down).
down). We
CANNOT conclude
conclude that
that
there
cause and
there is aa cause
relationship between
between the
two.
We
can’t
say
that
one
causes
the
other.
and effect relationship
the two . We can't say that one causes the other.
341
CHAYfER 30 ANSWERS
ANSWERS TO THE EXERCISES
CHAPTER
EXERCISES
Chapter29: Volume
Chapter
CHAPTER EXERCISE:
CHAPTER
EXERCISE:
1.
1.
0
: 7.5
5. \ 32.5 IThe width
width of the
the brick
brick is 6(1.25) =
inches, and
and its height
inches,
z 4 inches.
height is 6 -‐ 2 =
inches . The
brick's volume
brick’s
volume is then
then
lwh == (6)(7.5)(4)
inches. Since 1
1
lwh
(6)(7.5 )(4) =: 180 cubic inches.
grams, the brick's
kg = 1000 grams,
brick's mass
mass is 5,850
grams. Finally,
grams.
Finally, the
the brick’s
brick's density
density is
mass
_
mass _ 5,850
5,850 grams
volume
- - ‐180
1grams
in3
8 0 =‐ 32.5
32.5 grams
grams per
per
volume =
in3
cubic inch.
inch.
Each piece
half the
Each
piece is half
the cylinder.
cylinder.
_
V‐_12m 2h‐_127r(2)2(5) ‐107r
2. [!]The height
height of the
+ 225
z 44
the box is 100 ...;5=
(dividing
the volume
(dividing the
volume by
the base
base
by the
the area
area of the
gives us
us to the
the height).
gives
of
the
base
height). The sides
sides the
are
rectangular box
are x/E
v'25= 5 inches
inches long.
long . The rectangular
has dimensions
dimensions 5 x 5 x 4.
has
6. El
volume of
[illThis question
question is asking
asking for the volume
the cylinder.
cylinder . The
the
The radius
radius of the base
base is 2 and
and
since
tennis
ball
is
4,
the
since the diameter
diameter of each
each tenni s ball
height
3 = 12.
height of the
the cylinder
cylinder is 44 x 3
4
5
vV =:
5
= 487r
487T
7. ~ The shortest
shortest way
way to do this question
question is to
pretend
pretend that
liquefied and
that the block
block is liquefied
and poured
poured
into the aquarium.
aquarium. How
into
How high
level
high would
would the
the level
of the liquid
liquid rise?
and bottom
bottom have
have a
a surface
surface area
The top and
area of
2(5 x 5) = 50. The front and
have aa
and back have
surface
surface area
area of 2(5 x 4) = 40. The left
left and
and
right have
right
z 40.
have a
a surface
surface area
area of 2(5 x 4) =
The total
surface area
50 +
+ 40
40 +
+ 40
total surface
area is 50
40 = 130.
3.
7Tr2h
rrr 2 h =
= n(2)2(12)
rr(2)2 (12)
V = lwh
lwh
5. The cube
has
[f] Let the
the side
side of the cube
cube be
be s.
cube has
5,000 = (80)(25)h
(80) (25)11
2.5 =
: h
2.
six faces and
and the area
area of each
each face is s$2.
Solving
Solving for ss in terms
terms of a,
The longer
is to
to find
longer way
way to do this question
question is
the original
volume, add
the block,
original volume,
add the
block, find the
the
new
and then
then compare
compare it to the
new height,
height, and
original
o t the
original height.
height. While nnot
the fastest
fastest method,
method,
it is certainly
viable.
certain ly viab le.
652
6s2 = 24112
24a2
= 4112
s522 =
4a 2
s = 2a
2a
The volume
then 53=
volume is then
s3 = (2a)3 =: 8a3.
8.
4. E]
that can
can be
be filled in 3
[[] The cylindrical
cylindrical tank
tank that
2 (6) = 96rr.
hours has
hours
volume of
of rr(4)
7t(4)2(6)
967t. The
has aa volume
tank in question
tank
question has
has aa volume
volume of
7T(6)2(8)
Using the
tank as
asa
rr(6)2(8) = 28871.
288rr. Using
the first tank
a
conversion factor,
conversion
0
If
you take
the cubes
cubes with
with black
black
If you
take away
away all the
paint
them, you
essentially uncovering
uncovering
paint on them,
you are
are essentially
an
with a
an inner
inner cube
cube with
a side
side length
length of 3. A front
view is shown
shown below.
below .
.-- - -
I
I
hours
3 hours
28871 x --= 9 hours
2887T
hours
9671
967T
3
~
I
I
,_- - - .,
I
I
There
z 27 cubes
that are
are unpainted
unpainted..
There are 33
33 =
cubes that
342
THE COLLEGE
COLLEGE PANDA
THE
PANDA
9. ~ Since
Since each
each small
small cube
cube has
volume of
9.
has a
a volume
22‘ =
and the
= 8 and
the volume
volume of the
the outer
outer box
box is
3
=
there must
883
z 512, there
m u s t be
be 512 -i+8=
= 64 cubes
cubes in
the
that
box. If you
you take
take away
away all
all the
the cubes
cubes that
the box.
are
are touching
touching the
the box,
box, you
you are
are essentially
essentially
uncovering
uncovering an
an inner
inner rectangular
rectangular box
box with
with aa
square
square base
base of side
side 4 and
and aa height
height of 6. A
front
front view
view is shown
shown below.
below.
1
V
V = émzh
rrr 2h
3
6na 4
67m4
1
= -énr2(2a2)
rrr 2 (2a 2 )
3
2
187m44 =
2 rrr
7rr2(2a2)
18na
(2a 2 )
4
2 2
18na
=
r
187m4
= 2rra
27mzr2
9a22 =
: r2
9a
r2
8
3
3aa== r
I
61
I
I
4
I
•------~
12.
8
form a
a right
triangle with
and
to form
right triangle
with the
the radius
radius and
the slant
slant height.
triangle is
a multiple
height. This
This triangle
is a
multiple of
the
the
triangle: 9‐12‐15.
9-12-15. You
the 3--4-5
3‐4‐5 right
right triangle:
could've used
used the
could’ve
the pythagorean
pythagorean theorem
theorem
8
The
The volume
volume of this
this inner
inner rectangular
rectangular box
box is
is
4x4x6=
: 96. Since
Since each
each cube
cube has
has aa volume
volume
of 8, there
there are
cubes that
are 96...,...
96 + 8 =
= 12
12cubes
that are
are not
not
touching the
touching
the box,
box, which
which means
means there
there are
are
64
= 52
that are
64‐- 12
12=
52 cubes
cubes that
are touching.
touching. You
also
taken the
also could've
could’ve taken
the straight-forward
straight-forward
approach
approach of counting
counting up
up the
the cubes
cubes along
along the
the
took this
sides.
sides. If you
you took
this route,
route, you
you should've
should’ve
gotten
gotten something
something along
along the
the lines
lines of
1 6++16
1 6++88++88+
+ 4 = 52.
16
10.
instead
aware of this.
any
instead if
if you
you weren't
weren’t aware
this. In any
case,
the height
cone is 12
case, the
height of the
the cone
12cm.
cm.
volume of cone
V=
: volume
cone +
-+- volume
volume of hemisphere
hemisphere
_1!rrr 22h + !1(4
V‐37rrh+§
5m33)
)
=
(~rrr
3
2 3
V
~
2
3
v = ~n(9)
%n(9)2(12)
V
(12) + %(§n(9)3>
(;rr (9) )
[Q]
that have
IE]The
The only
only cubes
cubes that
have exactly
exactly one
one face
= 32471'
+ 486n
48671 =
= 81071
V =
3247T+
810n
painted
painted black
black are
are the
the ones
ones in the
the middle
middle of
each side
each
side.. For
For example,
example, the
the front
front side
side has
has
= 3 of these
these cubes.
cubes.
3x 1=
, _ - -
-
3
5
-
-
the radius
radius rr is 2 inches
longer than
13. ~ Since
Since the
inches longer
than
the
height is r -‐ 2.
Using the
2. Using
the
the height,
height , the
the height
volume formula
formula for aacylinder,
cylinder, we
get
volume
we get
2
2h =
V
7rr2h
: n7rr2(r
27rr2.
V = nr
r 2 (r -‐ 2) = 7rr3
nr 3 -‐ 2nr
.
3
:-------;1
the middle
middle of the
[I] Draw
Draw aa line
line down
down the
the cone
cone
_1
14. []] This
This question
asking for the
the
question is essentially
essentially asking
volume,
the amount
amount of rroom
volume, or the
o o m, in the
the crate.
crate.
The room
room in the
the crate
crate can
can be
as a
The
be seen
seen as
rectangular
with a
length of
rectangular box
box with
a length
1
c h e s ,aawidth
w i d t of
h o8f-8 ‐11- ‐ 11 =
: 66
=i 8n inches,
100-‐ 11 ‐- 11: 8
inches, and
and a height
inches.
inches,
height of 3 -‐ 1 = 2 inches.
4
The
The right
right side
side has
has 2 x 1 = 2 of these
these cubes,
cubes,
and
and the
the top
top has
has 3 x 2 =
= 6 of these
these cubes.
cubes. So
So
+ 2+
+6=
z 11 of these
these cubes.
cubes. To
far, we have
have 3 +
account
left, and
account for the
the back,
back, left,
and bottom
bottom sides,
sides,
we
we double
double this
this to get
get 22
22 cubes.
cubes.
6
V :=8 >
8 <X 66 >X<22=: 996
343
EXERCISES
CHAPTER
ANSWERS TO THE
THE EXERCISES
CHAPTER 30 ANSWERS
15.
blocks .
into 3 blocks.
vertically into
staircase vertically
Cut the staircase
[I] Cut
block 1
staircase =
Volume of staircase
= Volume of block
block 2
+
+ Volume of block
block 3
+
+ Volume of block
V=(5x2x0.2)+(5x2x0.4)+(5x2x0.6)
V = (5 X 2 X 0.2) + (5 X 2 X 0.4) + (5 X 2 X 0.6)
= 22 + 4 + 6
= 112
2
=
Density x Volume
Mass
= Density
Mass =
560 kg
= 130 x 12 =
= 1,
1,560
344