Combined and Joint Variation 129
Lesson
2-9
Combined and
Joint Variation
Lesson 2-9
BIG IDEA The same methods used to solve variation
problems involving two variables can be applied to variation
problems involving more than two variables.
Combined Variation
At the beginning of this chapter, you read about how adjusting the
number of teeth on the gears of a bicycle changes its speed. The
speed
S
of a bicycle varies directly with the number of revolutions
per minute (rpm)
R
that you turn the pedals and with the number
F
of teeth on the front gear. The speed also varies inversely with the
number
B
of teeth on the back gear. This situation is modeled by
the equation
S
=
kRF
__
B
.
This equation is read “
S
varies directly as
R
and
F
and inversely as
B
.”
When both direct and inverse variations occur together in a situation,
we say the situation is one of combined variation.
You saw another example of combined variation in Lesson 2-8, where
the maximum weight
M
of a board varied directly with its width w
and the square of its thickness
t
, and inversely with the distance
d
between its supports. This relationship was modeled by the equation
M
=
kwt
2
__
d
.
QY1
A combined-variation equation has two or more independent
variables, and the independent variables can have any positive
exponent. To fi nd
k
in a combined-variation model, use the same
strategy as in a variation problem with one independent variable:
• Find one instance that relates all the variables simultaneously.
• Substitute known values into the general variation equation.
• Solve for
k
.
Mental Math
Jeff is experimenting with
a balance scale. He nds
that 8 erasers balance
1 apple.
a. His calculator weighs
2.5 times as much as an
apple. How many erasers
does he need to balance
his calculator?
b. A pair of scissors weighs
the same as 2 erasers.
How many pairs of scissors
will balance the calculator?
c. There are two pairs of
scissors on one side of the
scale and the calculator
on the other. How many
erasers should he add to
the pan with the scissors
to balance the calculator?
Mental Math
Jeff is experimenting with
a balance scale. He nds
that 8 erasers balance
1 apple.
a. His calculator weighs
2.5 times as much as an
apple. How many erasers
does he need to balance
his calculator?
b. A pair of scissors weighs
the same as 2 erasers.
How many pairs of scissors
will balance the calculator?
c. There are two pairs of
scissors on one side of the
scale and the calculator
on the other. How many
erasers should he add to
the pan with the scissors
to balance the calculator?
QY1
Write an equation that
represents this statement:
y varies directly as the
square of x and inversely
as z.
QY1
Write an equation that
represents this statement:
y varies directly as the
square of x and inversely
as z.
Vocabulary
combined variation
joint variation
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130 Variation and Graphs
Chapter 2
Example 1
Mario is pedaling a bike at 80 revolutions per minute using a front gear
with 35 teeth and a back gear with 15 teeth. At these settings, he is
traveling 14.5 miles per hour. Describe how his speed would change if he
increased his pedaling to 100 rpm.
Solution
Use the combined variation equation S =
kRF
__
B
. Substitute
S = 14.5, R = 80, F = 35, and B = 15 and solve for k.
14.5 =
k(80)(35)
____
15
14.5 ≈ k(187)
0.078 ≈ k
So the variation formula for this situation is S =
0.078RF
____
B
.
To nd Mario’s new speed, substitute R = 100, F = 35, and B = 15 into
your formula and solve for S.
S =
0.078(100)(35)
_______
15
≈ 18
This means that by increasing his rpm to 100, Mario increases his
speed by about 3.5 mph (from 14.5 mph to 18 mph).
Another Example of Combined Variation
Photographers are always looking for ways to make
their pictures sharper. One way is to focus the lens at the
hyperfocal distance
. Focusing the lens at this distance
will produce a photograph with the maximum number of
objects in focus.
Photographers often use the hyperfocal distance when
taking pictures of landscapes. The two images at the right
were shot with the same camera at the same settings,
except the one on the bottom was taken with the lens
focused at the hyperfocal distance, giving the extra
degree of sharpness.
Example 2 explores how the hyperfocal length
H
can be
calculated using the focal length
L
of the camera lens in
millimeters and the f-stop, or aperture,
f
. The
aperture
is
a setting that tells you how wide the lens opening is on a
camera. It is the ratio of the focal length to the diameter
of the lens opening and so has no units.
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Combined and Joint Variation 131
Lesson 2-9
Example 2
The hyperfocal length H in meters varies directly with the square of the
focal length L in millimeters, and inversely with the selected aperture f.
a. Write a general variation equation to model this situation.
b. Find the value and unit of k when the hyperfocal length H is 10.42 m
when using a 50 mm lens (L) and the aperture f is set at 8.
c. Write a variation formula for the situation using your answer to Part b.
d. Find the hyperfocal length needed if you want to shoot with a 300 mm
lens at an aperture setting of 2.8. Include the units in your calculations.
Solution
a. Because the hyperfocal length varies directly with the square of the focal
length of the lens, L
2
will be in the numerator of the expression on the
right side of the formula. Because the hyperfocal length varies inversely
with the aperture setting, f will be in the denominator of the expression.
A general equation is H =
kL
2
__
f
.
b. Use your formula from Part a with: H = 10.42 m, L = 50 mm, and
f = 8. Include units when making substitutions. Here we show how to
do it by hand and with a CAS.
By hand:
10.42 m =
k(50 mm)
2
_____
8
Substitute the given
values for H, L, and f.
10.42 m =
2500 mm
2
·
k
______
8
Square 50 mm.
83.36 m = 2500 mm
2
· k Multiply both sides by 8.
0.033
m
__
mm
2
≈ k Divide both sides by 2500 mm
2
.
c. Substitute k into the equation from Part a.
H =
0.033 L
2
____
f
d. Use the formula from Part c to calculate H when L = 300 mm and
f = 2.8. Include the units when you substitute.
H =
0.033
m
__
mm
2
(300 mm)
2
_________
2.8
H =
0.033
m
__
mm
2
(90,000 mm
2
)
__________
2.8
H ≈ 1060.7 m
The hyperfocal length is about 1,060 meters.
With a CAS:
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132 Variation and Graphs
Chapter 2
Joint Variation
Sometimes one quantity varies directly as powers of two or more
independent variables, but not inversely as any variable. This is
called joint variation. The simplest joint-variation equation is
y
=
kxz
,
where
k
is the constant of variation. The equation is read “
y
varies
jointly as
x
and
z
” or “
y
varies directly as the product of
x
and
z
.”
Guided example 3 explores a joint-variation situation in geometry.
Example 3
The volume of a solid with a circular base varies jointly as the height of the
solid and the square of the radius of the base.
a. Write a general equation to model this situation.
b. If the volume of the solid is approximately 75.4 cubic centimeters when
the radius is 2 centimeters and the height is 6 centimeters, nd the
value of k.
c. The value of k is approximately equal to what famous mathematical
value?
d. Write a variation formula using your answer to Part b.
e. What well-known kind of solid gure could this be?
Solution
a. Let V be the volume of the solid, h be the height of the solid, and r be
the radius of the base.
A general equation is V = k
?
.
b. Use your formula from Part a with V = 75.4, h = 6, and r = 2 to solve
for k.
75.4 = k
?
?
2
75.4 = k
?
?
≈ k
c. The value of k is approximately equal to
?
.
d. Substitute k into your equation from Part a. So, V =
?
.
e. Based on the formula in Part d,
this solid could be a(n)
?
.
Many other geometry formulas can be interpreted as direct- or joint-
variation equations.
QY2
GUIDEDGUIDED
QY2
Translate this statement
into an equation: The area
of a triangle varies jointly
with the height and the
base of the triangle. What
is the constant of variation
for this formula?
QY2
Translate this statement
into an equation: The area
of a triangle varies jointly
with the height and the
base of the triangle. What
is the constant of variation
for this formula?
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Combined and Joint Variation 133
Lesson 2-9
As in combined variation, a joint-variation equation can have more
than two independent variables, and the independent variables can
have any positive exponent.
Questions
COVERING THE IDEAS
1. a.
What is combined variation?
b. How is joint variation different from combined variation?
2. Translate into a single formula:
M
varies directly as
t
and
r
2
and
inversely as
d
.
3
.
Refer to Example 1. Suppose Mario slowed his pedaling to
75 rpm. What would his speed be?
4. In Example 2, assume that a photographer calculated a
hyperfocal length of 6.54 meters. If he used a lens with a focal
length of 28 millimeters, at what aperture was the lens set?
5. Sonia calculated the volume of a cylindrical can of cat food with
a radius of 1.5 inches and a height of 1.2 inches to be about
9.2 cubic inches.
a. Use the equation
V
=
kr
2
h
to calculate the constant of
variation
k
that she used.
b. Did Sonia use the correct formula? How do you know?
6. Refer to Example 3. If the volume of the solid is 25.13 cm
3
with a
radius of 2 cm, what is its height?
APPLYING THE MATHEMATICS
7.
The formula
F
=
ma
gives the force
F
on an object with mass
m
and acceleration
a
.
a. Rewrite the formula in words using the language of variation.
b. What is the constant of variation?
8. The volume of a solid with a circular base varies jointly with the
square of the base radius and the height. When the volume is
83.78 cm
3
, the height is 5 cm and the base radius is 4 cm.
a. Write a general equation to model this situation and fi nd
k
.
b.
k
is a multiple of π. Write
k
in terms of π.
c. Write a variation formula for the volume of this solid in terms
of π.
d. What kind of well-known solid is this?
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134 Variation and Graphs
Chapter 2
9. The volume of a certain solid varies jointly as its height, width,
and length.
a. Write a general equation to model this situation.
b. Fill in the Blank The answer to Part a suggests that this is a
?
solid.
c. Based on your answer to Part b, fi nd the value of
k
.
10. Suppose
y
varies directly as
x
and inversely as
z
. Describe how
y
changes when
x
and
z
are each tripled. Explain your answer.
11. The cost
C
of polyvinyl chloride (PVC) piping in dollars varies
jointly as the length
L
of the pipe and the difference between the
squares of its outer and inner radii,
R
o
2
-
R
i
2
. Suppose that a
foot of PVC piping with an outer radius of 0.25 foot and an inner
radius of 0.20 foot costs $3.72.
a. Using the given variables, write a joint variation equation.
b. Find the constant of variation.
c. Rewrite the variation equation using the constant from Part b.
d. Find the cost of 10 feet of PVC piping with an outer radius of
0.5 foot and in inner radius of 0.48 foot.
e. Determine the unit of the constant of variation.
REVIEW
12.
The Ideal Gas Law in chemistry relates the pressure
P
(in
atmospheres) exerted by a gas to the temperature
T
(in Kelvins)
of the gas and volume
V (
in liters) of its container. Chantel
obtained the following data using a 5-liter container in the lab.
(Lessons 2-7, 2-8)
T
(K)
235 260 285 305
500
P
(atm)
0.7285 0.8060 0.8835 0.9455
1.55
a. Graph the data points.
b. How does
P
vary with
T
?
c. With a temperature of 350 Kelvins, Chantel manipulated the
volume of the container and obtained the following data.
Graph these points on a different set of axes.
V
(L)
1234
56
P
(atm)
5.425 2.713 1.808 1.356
1.085 0.603
d. How does
P
vary with
V
?
e. Write an equation that relates
P
,
T
, and
V
. You do not need to
fi nd the constant of variation.
R
o
R
i
L
R
o
R
i
L
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Combined and Joint Variation 135
Lesson 2-9
13. Let
g
:
x
→ 3
x
2
. (Lesson 2-5)
a. Graph
g
over the domain
–
2.5 ≤
x
≤ 2.5.
b. What is the name of the shape of this graph?
c. Find the average rate of change of
g
between
x
= 1 and
x
= 2.
d. Would the answer to Part c be the same for any two values of
x
that differ by 1? Explain your answer.
14. If
y
varies directly as
w
3
, and
y
= 25 when
w
= 5, fi nd the value
of
y
when
w
= 2. (Lesson 2-1)
15. One general equation for a combined variation is
y
=
k
xz
_
w
.
Solve for
k
in terms of the other variables. (Lesson 1-7)
16. When Clara tried to solve the equation
1
_
5
x
+
1
_
7
x
+ 2 = 5, her fi rst
step led to 7
x
+ 5
x
+ 70 = 175. (Lesson 1-6)
a. Explain what Clara did.
b. Finish solving the equation.
17. Given the function
f
:
x
→ 4
x
3
- 2
x
+ 1, fi nd
f
(π). (Lesson 1-3)
EXPLORATION
18.
There are many instances of mathematics on TV shows. Watch
a show in which mathematics plays a role and record any
mathematics used on the show. Note whether the mathematics
was used accurately and try to explain any mistakes.
QY ANSWERS
1. y =
kx
2
__
z
2. A = kbh. In this case,
k =
1
_
2
.
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