Chabot Mathematics
§6.8 Model
by Variation
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Review § 6.7
MTH 55
 Any QUESTIONS About
• §6.7 → Formulas and Applications of
Rational Equations
 Any QUESTIONS About HomeWork
• §6.7 → HW-28
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
§6.8 Direct and Inverse Variation
 Equations of
Direct Variation
 Problem Solving
with Direction Variation
 Equations of Inverse Variation
 Problem Solving with
Inverse Variation
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Direct Variation
 Many problems lead to equations of the
form y = kx, where k is a constant. Such
eqns are called equations of variation
 DIRECT VARIATION →
When a situation translates to an
equation described by y = kx, with k a
constant, we say that y varies directly
as x. The equation y = kx is called an
equation of direct variation.
Chabot College Mathematics
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BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Variation Terminology
 Note that for k > 0, any equation of the
form y = kx indicates that as x
increases, y increases as well
 Synonyms
• “y varies as x,”
• “y is directly proportional to x,”
• “y is proportional to x”
 The Synonym Terms also imply direct
variation and are often used
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BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
The Constant “k”
 For the Direct
Variation Equation
y  kx
 The constant k is called the
constant of proportionality or the
variation constant.
 k can be found if one pair of values
for x and y is known.
 Once k is known, other (x,y) pairs can
be determined
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Direct Variation
 If y varies directly
 Solving for k
as x, and y = 3 when
3 1
k    0.25
x = 12, then find the
12 4
eqn of variation
 Thus the Equation
 SOLUTION: The
of Variation
words “y varies
directly as x”
1
y    x  0.25 x
indicate an equation
4
of the form y = kx:
y  kx  3  k  12
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Direct Variation cont.
 Graphing the
Equation of
Variation
x
5
2
.
y0
Chabot College Mathematics
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 Direct Variation
Always
produces a
SLANTED LINE
that Passes
Thru the
ORIGIN
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Direct Variation
 Find an equation in
which a varies
directly as b, and
a = 15 when b = 25.
 Thus the Variation Eqn
a  0.6b
 Sub b = 36 into Eqn
3
a  0.636  21.6  21
5
 Find the value of a
when b = 36
 SOLUTION:
a  kb  15  k  25
15 3
k
  0.6
25 5
Chabot College Mathematics
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 Thus when b = 36,
then the value of a
is 21-3/5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Bolt Production

The number of bolts B that a machine
can make varies directly as the time
T that it operates.
 The machine makes 3288 bolts in 2 hr
 How many bolts can it make in 5 hr
1. Familarize and Translate: The
problem states that we have DIRECT
VARIATION between B and T.
•
Thus an equation B = kT applies
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Bolt Production cont.1
3. Carry Out: B  kT  3288bolts  k  2hr
3288bolts
bolts
• Solve for k: k 
 1644
2hr
hr
bolts 

• Thus the Equation
B  1644
T

of Variation:
hr 

•
If T = 5 hrs:
bolts 

B  1644
  5hr  8220bolts
hr 

•
Note that k is a RATE with UNITS
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Fluid Statics
 The pressure exerted by a liquid at
given point varies directly as the
depth of the point beneath the surface
of the liquid.
 If a certain liquid exerts a pressure of 50
pounds per square foot (psf) at a depth
of 10 feet, then find the pressure at a
depth of 40 feet.
 SOLN: Another case of Direct Variation
Chabot College Mathematics
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BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Fluid Statics
Chabot College Mathematics
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(units are lb/ft3)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Fluid Statics
 Use k = 5 lb/ft3 in the Direct Variation
Equation to find the pressure at a depth
of 40ft
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Inverse Variation
 INVERSE VARIATION →
When a situation translates to an
equation described by y = k/x, with k a
constant, we Say that y varies
INVERSELY as x. The equation
y = k/x is called an equation of
inverse variation
• Note that for k > 0, any equation of the form y =
k/x indicates that as x increases, y decreases
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Inverse Variation
 If y varies inversely
as x, and y = 30
when x = 20, find
the eqn of variation
 SOLUTION: The
words “y varies
inversely as x”
indicate an equation
of the form y = k/x:
k
k
y   30 
20
x
Chabot College Mathematics
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 Solving for k
k  30  20  600
 Thus the Equation
of Variation
600
y
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Barn Building

It takes 56 hours for 25 people
to raise a barn.
How long would it
take 35 people to
complete the job?

•
Assume that all
people are
working at the
same rate.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Barn Building cont.1
1. Familarize. Think about the situation.
What kind of variation applies? It
seems reasonable that the greater
number of people working on a job,
the less time it will take. So LET:

•
T ≡ the time to complete the job, in hours,
•
N ≡ the number of people working
Then as N increases, T decreases and
inverse variation applies
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Barn Building cont.2
2. Translate: Since inverse
k
T

variation applies use
N
3. Carry Out: Find the Constant of
Inverse Proportionality
k
T
N
k
56hrs 
25workers
k  56  25  1008 Worker  Hrs
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Barn Building cont.3
k
3. Carry Out: The
T

Eqn of Variation
N
 When N = 35. Find T
1008worker • hrs
T
35 workers
T  28.8hrs  29hrs
4. Chk: A check might be done by
repeating the computations or by
noting that (28.8)(35) and (56)(25)
are both 1008.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Barn Building cont.4
5. STATE: if It takes 56 hours for 25
people to raise a barn, then it should
take 35 people
about 29 hours
to build the
same barn
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
To Solve Variation Problems
1. Determine from the language of the
problem whether direct or inverse
variation applies.
2. Using an equation of the form y = kx
for direct variation or y = k/x for
inverse variation, substitute known
values and solve for k.
3. Write the equation of variation and use
it, as needed, to find unknown values.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Applications Tips ReDux
 The Most Important Part of Solving
REAL WORLD (Applied Math) Problems
 The Two Keys to the Translation
• Use the LET Statement to ASSIGN
VARIABLES (Letters) to Unknown Quantities
• Analyze the RELATIONSHIP Among the
Variables and Constraints (Constants)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Solving Variation Problems
1. Write the equation with the constant of
variation, k.
2. Substitute the given values of the
variables into the equation in Step 1 to
find the value of the constant k.
3. Rewrite the equation in Step 1 with the
value k from Step 2
4. Use the equation from Step 3 to
answer the question posed in the
problem.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Other Variation Relations
 Some Additional Variation Eqns:
 y varies directly as the nth power of x
if there is some positive y  kx n
constant k such that
 y varies inversely as the nth power of
x if there is some positive y  k
n
x
constant k such that
 y varies jointly as x and z if there is
some positive constant k y  kxz
such that.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Combined Variation
 The Previous Variation Forms can be
combined to create additional equations
 z varies directly as x and
x
INversely as y if there is some z  k
y
positive constant k such that
 w varies jointly as x & y and
inversely as z to the nth
wk
power if there is some positive
constant k such that
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
xy
n
z
Example  Luminance
 The Luminance of a light (E) varies
directly with the intensity (I) of the light
source and inversely with the square
distance (D) from the light. At a distance
of 10 feet, a light meter reads 3 units for
a 50-cd lamp. Find the Luminance of a
27-cd lamp at a distance of 9 feet.
 This is a case of
I
COMBINED Variation → E  k 2
Chabot College Mathematics
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D
k  50
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Luminance
 Solve for the
Variation Constant, k,
Using the KNOWN
values of I & D
I
Ek 2
D
k  50
3
2
10
6k
 Use the value of k, and
6  27
E 2
D = 9ft in the variation
9
eqn to find E(9ft)
E2
 State: At 9ft the 27cd Lamp produces
a Luminance of 2 cd/m2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Sphere Volume
 Suppose that you had forgotten the
formula for the volume of a sphere, but
were told that the volume V of a sphere
varies directly as the cube of its
radius r. In addition, you are given that
V = 972π when r = 9in.
 Find V when r = 6in
 SOLUTION: Recognize as Direct
Variation to a Power: V = kr3
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Sphere Volume
 Now use KNOWN
data to solve for k
 Now Substitute
k = 4π/3 into the
Eqn of Variation
4 3
V  r
3
Chabot College Mathematics
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V  kr 3
972  k 9 
3
972  k 729 
972
k
729
4
k 
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Sphere Volume
 Finally Substitute r = 6 and solve for
V(6) as requested
4
3
V   6 
3
 288 cubic inches
 Using π ≈ 3.14159 find the Volume for a
6 inch radius sphere, V(6) ≈ 904.78 in3
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Newton’s Law
 Newton’s Law of Universal Gravitation
says that every object in the universe
attracts every other object with a force
acting along the line of the centers of the
two objects and that this attracting force
is directly proportional to the product
of the two masses and inversely
proportional to the square of the
distance between the two objects.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Newton’s Law
 Write the Gravitation Law Symbolically
 SOLUTION: Let m1 and m2 be the
masses of the two objects and r be the
distance between them; a Diagram:
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Newton’s Law
 Next LET:
• F ≡ the gravitational force between the
objects
• G ≡ the Constant of Variation; a.k.a., the
constant of proportionality
 Thus Newton Gravitation Law in
Symbolic form
m1m2
F  G 2 .
r
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Newton’s Law
 The constant of proportionality G is called the
universal gravitational constant. It is termed a
“universal constant” because it is thought to
be the same at all places and all times and
thus it universally characterizes the intrinsic
strength of the gravitational force.
 If the masses m1 and m2 are measured in
kilograms, r is measured in meters, and the
force F is measured in newtons, then the
value of G:
3
G  6.67 10
Chabot College Mathematics
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11
m
2
kg  s
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Newton’s Law


Next Estimate the value of g; the
“acceleration due to gravity” near the
surface of the Earth. Use these
estimates:
•
Radius of Earth RE = 6.38 x 106 meters
•
Mass of the Earth ME = 5.98 x 1034 kg
SOLUTION: By Newton’s 1st Law
Force = (mass)·(acceleration) →
F  ma
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Newton’s Law

Now the “Force of Gravity” at the
earth’s surface is the result
F  m g
of the Acceleration of Gravity:
 Equating the “Force of Gravity” and the
Gravitation Force Equations:
m1  m2
F  G

m

g

F
2
r
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Newton’s Law

Carry
mM E
mg  G
Out
R 2
E
ME
g  G 2
RE
6.67  10  5.98  10 

g
6.38  10 
11
24
6 2
g  9.8 m/sec 2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Kinetic Energy
 The kinetic energy of an object varies
directly as the square of its velocity.
 If an object with a velocity of 24 meters
per second has a kinetic energy of
19,200 joules, what is the velocity of an
object with a kinetic energy of 76,800
joules?
 SOLUTION: This is case of Direct
Variation to the Power of 2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Kinetic Energy
 Write the Equation of Variation
 Next Solve for the Variation Constant, k,
using the known data
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Kinetic Energy
 To find k, use the fact that an object
with a velocity of 24 m/s has a kinetic
energy of 19.2 kJ
 Thus k = 33.33 J/m2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Kinetic Energy
 Use k = 33.33 J/m2 to refine the
Variation Equation
 Next use the E(v) eqn to find v for
E = 76.8 kJ2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Example  Kinetic Energy
 The v for E = 76.8 kJ
 Thus when E = 76.8 kJ the velocity
is 48 m/s
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
WhiteBoard Work
 Problems From §6.8 Exercise Set
• 33, 38

KINETIC and
POTENTIAL
Energy
Balance
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
All Done for Today
Heat Flows
Hot→Cold
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Chabot College Mathematics
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5
y = |x |
6
5
4
3
2
1
0
1
2
3
4
5
6
y
4
3
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
file =XY_Plot_0211.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
4
5
6
y
5
2
y
1
x
4
0
-6
-5
-4
-3
-2
-1
3
0
1
2
3
4
-1
-2
2
-3
1
-4
x
-5
5
-6
0
-3
-2
-1
0
1
2
3
-1
4
-7
-8
-2
-9
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
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file =XY_Plot_0211.xls
-10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt
5
6