Algebra II
Rational Functions
1) Pure water is mixed with 20 milliliters of 25% acid solution.
a) How many milliliters of acid are in the original solution?
b) If you add 4 milliliters of pure water to your original solution, what is the percent of
acid in the new concentration? How did you find this?
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c)
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If your original 20 milliliters of 25% acid is change to a solution of 10% acid, how
many milliliters of pure water were added to obtain this concentration?
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d) Write a function for the concentration, C, of acid in the mixture in terms of the
amount of, x, of pure water added.
e) Graph the concentration function, and explain what the behavior of the graph
represents.
~1~t1 ~l~t~ ~1~t3
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....
Y1B5/(20+X
XNin=0
Xr·'.ax=50
Xscl=10
Yr')in=0
Yr·)ax=.5
Yscl=.25
Xres=l
....
Y~=
....
Y3=
....
Ylj=
....
Y~=
....
Y6=
....
Y7=
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2) The concentration C of a certain drug in a patient's bloodstream t minutes after injection
is given by
C(t) =
?
(r
30t
+ 35)
a) Graph this function.
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XJYlin=0
Xr..•
ax=30
Xscl=5
Yr'lin=0
YJYlax=5
Yscl=l
Xres=1
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lA+
+,''''''
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c) What happens to the concentration of the drug as the time increases?
~e
C-CV\Cl~~+''o",
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y ~ ~
1~C.-ICctSt)
,C\fiJ
o1.ac,c'tSt'~
slclAIY.
d) What is (are) the asymptote(s) for this function.
asymptote( s).
j =- 0
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I )
Y't'\,"~1'\
'"l<!lCJ
hcri ~-l- (
0..
~
O\~
tk..+
+;"'\~
~(
01.
CCJ'oI\
Explain the meaning of the
tj rhA-~.
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So
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qr p
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e) What is the domain and range of this situation?
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Explain.
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x?- C> ~
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3) If a store owner wants to sell an item for $200.00, he can mark the regular price of the
item as $250.00, which is a 25% increase and then sell it at a 20% discount of the
regular price. In other words, the store owner increases the price by a certain percent
and then advertises it at a discounted percentage to return the price to the original
desired price.
a) Show that taking a $200.00 item and increasing the price by 25% to obtain the
regular price then reducing the regular price by 20% will give the return the sale
price back to the desired $200.00.
L 0 0 -t 0:2. 5 ( 200) _ 1."2.5 ( 2.GO) ::- 2. SO'
250 -
o. '2..(2-5c)
::. O. ~ (2.50)
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( Y'l D1',c..(
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I
)
0, %
::. 2.00
b) If the store owner increases the $200.00 price by 15%, how much of a discount
should he/she advertise the item for in order to sell it at $200.00?
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2-00
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1.3
c) If the store owner wants to advertise the item for sale at a 40% discount and still sell
it for $200.00, what percent should he/she markup the price before the sale?
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~t
l'Ylo.rked
f';c.e,
Mo.rK"
I
if selllJ
vr ~ Fe.c+cr-
ct1e-f-
o,{g
o~(p
of
=. ~ •
d) Write a function d(m) for this situation where m is the percentage of markup and
d(m) is the sale percentage discount, always returning the price to the one the store
cA (M)
owner originally planned.
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t'\v
lec.~ffl>C-6<1
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Ih fl ic.e...·ko"\
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f;r
i';)
f\ 0
t
fcrc..e:--.-h..,.~
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-=
100 (
I-
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IGt>
t\ e
-
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I 00 + rV\
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R""
-r ~
r,.",I+;r"·c'\.-h·co-t ~-kr
ft'CU'~
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e) Explain what the behavior of the graph represents. Include the domain and ran e
for your function and any asymptotes it may have.
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4) The average cost function is defined as:
C(x)
=
C(x)
x
The cost function C (in thousands of dollars) for printing x textbooks (in thousands of
units) was found to be C(x) O.25x3 + O.58x2 + 9x + 78.5
=
0. Z-S.,e.? -rD. Sl5'/-'
-f-
cr~ + 7?5.S
X
tl~t1 tl~t~ tl~t3
~Y1a•.25XA3+.58X
~+9X+78.5)/X
~Y~=
~Y3=
~YIt=
~Ys:=
~Y6=
WI~mow
v
Xr lin=0
Xr lax=40
f
f
Xscl=l
Yp)in=0
Yf')ax=100
Yscl=l
Xt"es=l
d) Find the number of textbooks that should be printed to minimize average cost. What
is the minimum average cost?
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5) The United States Post Office has contacted you to design a closed box with a square
base that has a volume of 5,000 cubic inches.
a) Draw a picture of the situation
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\ _ 500C>
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b) Does the height have to be the same dimension as a side of the square base?
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c) Find a function for the surface area of the box
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-to
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S~~) ";.
z.
2 y..+-
'2.o00c;
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d) What are the dimensions of the box that minimize the surface area?
WINDOW
Xl"')in=0
Xl"')ax=40
Xscl=l
Yr.-.
i n= -5000
Yl"')ax=15000
Yscl=l
Xres=1
e) Why would the post office want to minimize the surface area of the box?
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6) A company manufactures aluminum cans in the shape of a cylinder with a capacity of
250 cubic centimeters. The top and bottom of the can are made of an alloy that costs
0.04 cents per square centimeter. The sides of the can are made of material that costs
0.03cents per square centimeter.
a) Express the cost of material for the can as a function of the radius r of the can
Cost ~ o,O'"ll
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+O.o3(
c..((!"\
of 5l'cAes)
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+- 0 ,03( ZiTr ~')
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-= 0.08 'if,? ..•..o,OC> 'if,.. ~
'If -;:..250
-=-=.
0 ,O?)((~ L -to O,Ofp 'iT. I ~)
l
O,O~71r~
1"
~
1(r 2-
:: 'iTr'2..h
II
~ ~ 2.5~
tr( "l..
r
b) Find the least cost
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Xr..•
in=0
XMax=10
Xscl=l
YMin=0
YMax=20
Yscl=l
Xres=l
~e
Mil"liro"Juro"J
X=3.1017S'13 .••••=7.;::S:~91i3B•
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