Evaluate: Homework and Practice
EVALUATE
ASSIGNMENT GUIDE
Practice
Explore Activity
Investigating Inverse Variation
Exercises 1–2
Example 1
Formulating and Solving Inverse
Variation Equations
Exercises 3–12
Example 2
Distinguishing between Inverse
Variation and Direct Variation
Exercises 13–23
Let the area of a rectangle be 16 square units. On the coordinate plane shown, draw
all rectangles having positive-integer lengths and widths. (Note that it doesn’t matter
whether the width is less than, equal to, or greater than the length.) Each rectangle
should have its lower-left corner at the origin. The first rectangle, having a length of
1 and a width of 16, is already drawn for you. After drawing the rectangles, draw a
smooth curve through their upper-right corners.
2.
Write an equation of the curve that you drew in
Evaluate 1. The equation should give the width
w of a rectangle in terms of the length ℓ.
w=
16
_
ℓ
Width
Concept and Skills
1.
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• Online Homework
• Hints and Help
• Extra Practice
w
ℓ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Length
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
3.
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Have students consider an inverse variation in which
the length of a rectangle is based on the area of the
rectangle in square feet being a constant product of
the length and width in feet, but then all units are
changed to inches. An inverse variation model can
still be used, but the change in units will result in a
change in the inverse variation constant.
Write an equation relating the variables and use it to answer the question.
Given that y varies inversely as x, and y = 8 when x = 3, what is the value of y
when x = 12?
Find the value of the constant of variation, a.
a
a
y= x 8=
24 = a
3
24
So, an equation relating x and y is y = x .
24
When x = 12, y =
= 2.
12
Given that y varies inversely as x, and y = 10 when x = 4, what is the value of x when
y = 8?
_
_
_
4.
_
Find the value of the constant of variation, a.
a
a
y = x 10 =
40 = a
4
40
So, an equation relating x and y is y = x . Using the equivalent equation
xy = 40, substitute 8 for y and solve for x to get x = 5.
_
_
_
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Exercise
461
Lesson 9.1
Lesson 1
461
Depth of Knowledge (D.O.K.)
Mathematical Processes
1–2
1 Recall of Information
1.E Create and use representations
3–4
2 Skills/Concepts
1.E Create and use representations
5–12
2 Skills/Concepts
1.A Everyday life
13–22
2 Skills/Concepts
1.E Create and use representations
23
2 Skills/Concepts
1.A Everyday life
24
3 Strategic Thinking
1.G Explain and justify arguments
2/21/14 12:30 AM
5.
The time t (in hours) that it takes a pump to empty a tank of water varies inversely
with the pumping rate r (in gallons per hour). If it takes 3 hours to empty a tank of
water when the pumping rate is 80 gallons per hour, how long does it take to empty
the tank when the pumping rate is 60 gallons per hour?
AVOID COMMON ERRORS
When graphing an inverse variation function without
real-world restrictions, students may graph only the
branch of the function in the first quadrant. Remind
students that negative numbers are included in the
domain, and that k can be represented by the product
of two negative numbers. For example, for
6 , (-2, -3) is a solution.
y = __
x
Find the value of the constant of variation, a.
a
a
240 = a
t= r 3=
80
240
240
So, an equation relating t and r is t = r . When r = 60, t =
= 4.
60
So, it takes 4 hours to empty the tank when the pumping rate is 60 gallons per hour.
_
6.
_
_
_
The number of flowers f that a gardener can plant along a border of a garden varies
inversely with the distance d (in inches) between the flowers. If the gardener can fill
the border with 30 flowers planted 12 inches apart, how far apart should the gardener
plant 36 flowers?
a
a
f=
30 =
360 = a
12
d
360
So, an equation relating f and d is f =
. Using the equivalent equation fd = 360,
d
substitute 36 for f and solve for d to get d = 10.
_
_
_
So, the gardener can plant 36 flowers 10 inches apart.
7.
The number of presents p that Tim can afford to buy varies inversely with their
average cost C (in dollars). If Tim can afford 5 presents when their average cost is $12,
what average cost would 3 presents have?
a
a
p=
60 = a
5=
12
C
60
So, an equation relating p and C is p =
.
C
Using the equivalent equation Cp = 60, substitute 3 for p and solve for C to get C = 20.
_
_
_
So, 3 presents would have an average cost of $20.
8.
_
_
© Houghton Mifflin Harcourt Publishing Company
A club rents a bus for a trip. The cost C (in dollars) that each person pays to cover the
cost of the bus varies inversely with the number of people p who go on the trip. It will
cost $30 per person if 50 people go on the trip. How much will it cost per person if
40 people go on the trip?
a
a
C = p 30 =
1500 = a
50
1500
1500
So, an equation relating C and p is C = p . When p = 40, C =
= 37.5.
40
So, it will cost $37.50 per person if 40 people go on the trip.
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Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
25
3 Strategic Thinking
1.F Analyze relationships
26
3 Strategic Thinking
1.A Everyday life
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Inverse Variation 462
9.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
For a fundraiser, members of the booster club wash cars by hand. The time t (in
minutes) it takes to wash a car varies inversely as the number of people p who are
washing the car. If 2 people can wash a car in 20 minutes, how many people would
be needed to wash a car in 8 minutes?
a
a
40 = a
t = p 20 =
2
40
So, an equation relating t and p is t = p . Using the equivalent equation
pt = 40, substitute 8 for t and solve for p to get p = 5.
So, 5 people are needed to wash a car in 8 minutes.
_
When determining whether an inverse variation can
be used to model a situation, students should
recognize that a value of 0 for either the dependent or
independent variable indicates that an inverse
variation is not an appropriate model.
_
_
10. A gear with 32 teeth meshes with a gear with 40 teeth so that when one gear revolves,
the other one does as well. The number of revolutions r that each gear makes varies
inversely with the gear’s number of teeth t. When the gear with 32 teeth makes
5 revolutions, how many revolutions does the gear with 40 teeth make?
a
a
5=
160 = a
r=
t
32
160
160
So, an equation relating r and t is r = ___
. When t = 40, r = ___
= 4.
t
40
_
_
So, the gear with 40 teeth makes 4 revolutions.
11. Music The frequency f (in hertz) of a vibrating guitar string varies inversely as
its length ℓ (in centimeters). If a guitar string 65 centimeters long vibrates with a
frequency of 110 hertz, at what frequency would the guitar string vibrate when the
guitarist reduces the string’s length to 22 centimeters?
a
a
f=
7150 = a
110 =
65
ℓ
7150
7150
So, an equation relating f and ℓ is f =
. When ℓ = 22, f =
= 325.
22
ℓ
So, the string vibrates with a frequency of 325 hertz when the string length is 22 centimeters.
© Houghton Mifflin Harcourt Publishing Company
_
_
_a
_
a
100 = a
50 =
2
d
100
100
So, an equation relating m and d is m =
. When d = 5, m =
= 20.
d
d
So, a force of 20 pounds applied at a distance of 5 feet from the fulcrum counterbalances
the force of 50 pounds applied at a distance of 2 feet from the fulcrum.
m=
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Lesson 9.1
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12. Physics When a lever is placed on a fulcrum and a force is applied at each end, the
lever will be in balance as long as the magnitude m (in pounds) of each force and
the distance d (in feet) of each force from the fulcrum satisfy an inverse variation
relationship. If a force of 50 pounds is applied to one end of a lever at a distance of
2 feet from the fulcrum, what force must be applied to other end, which is 5 feet from
the fulcrum, to bring the lever into balance?
Module 9
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463
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Lesson 1
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Determine whether the two variables vary inversely or directly.
Then write an equation and use it to answer the question.
13. Given the table of data, what is y when x = 15?
Because the products xy are 30, y varies
30
inversely as x, and y = __
x.
x
1
3
4
y
30
10
7.5
x
2
5
6
y
30
75
90
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Technology
When generating a table for an inverse variation, an
error should be expected when x = 0, since the
inverse variation function is undefined due to
division by 0. When tracing along the curve of an
inverse variation function, there should be no value
of the function when x = 0.
30
When x = 15, x = __
= 2.
15
14. Given the table of data, what is y when x = 10?
Because the ratios _x are 15, y varies directly
y
as x, and y = 15x.
When x = 10, y = 15(10) = 150.
15. The table gives the cost (in dollars) per person
when friends share in renting a mountain cabin
for a weekend. What is the cost per person
when 6 friends rent the cabin?
Number of People, p
2
3
5
Cost per Person, C
90
60
36
180
Because the products pC are 180, C varies inversely as p, and C = ___
p .
180
When p = 6, C = ___
= 30, so the cost per person when 6 friends rent the cabin is $30.
6
16. The table gives the amount of gas (in gallons) used
when driving a car various distances (in miles)
on highways. What amount of gas is used when
driving 336 miles on highways?
Distance Driven, d
112
140
224
Amount of Gas, g
4
5
8
d
1
are __
, g varies directly as d, and g = __
.
Because the ratios _
28
28
d
g
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Elisseeva/Shutterstock
336
When d = 336, g = ___
= 12, so 12 gallons of gas are used when driving 336 miles on
28
highways.
17. The table gives the cost (in dollars) of renting a rowboat at a
lake for various amounts of time (in hours). What is the cost of
renting a rowboat for 3.5 hours?
Time, t
2
2.5
3
Rental Cost, C
28
35
42
C
Because the ratios _
t are 14, C varies directly as t, and C = 14t.
When t = 3.5, C = 14(3.5) = 49, so the cost of renting a rowboat for 3.5 hours is $49.
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Inverse Variation 464
18. The table gives the total working time (in hours) that it takes a crew of painters to paint a house.
How much time does it take 6 painters to paint the house?
MODELING
Students should be aware that data is not always
modeled perfectly by a function, especially when it is
approximate or rounded. An inverse variation
function can still be used when the product of the
variables is not exactly constant. For example, when
the products are 37.488, 37.504, 37.496, and so on, an
inverse variation model with a value of k of 37.5 is
appropriate.
Number of Painters, p
2
3
5
Total Working Time, t
48
32
19.2
96
Because the products pt are 96, t varies inversely as p, and t = __
p.
96
When p = 6, t = __
= 16, so 6 painters can paint the house in 16 hours.
6
19. The table gives the speed of a bicycle (in miles per hour) when a cyclist pedals at various rates (in
revolutions per minute of the pedals) with the bicycle in a particular gear. What is the bicycle’s speed
when the cyclist pedals at a rate of 72 revolutions per minute?
Pedaling Rate, r
30
60
90
Bicycle’s Speed, s
5
10
15
1
r
Because the ratios _sr are _
, s varies directly as r, and s = _
.
6
6
72
__
When r = 72, s = = 12, so the bicycle’s speed is 12 miles per hour when the cyclist pedals
6
at a rate of 72 revolutions per minute.
20. The table gives the number of laptops sold in a month when a store sells a particular model of laptop at various
prices (in dollars). How many laptops would be sold in a month when the store sells the laptop for $500?
Price of Laptop, p
600
720
800
Number of Laptops Sold, ℓ
60
50
45
Because the products pℓ are 36,000, ℓ varies inversely as p, and ℓ = _____
p .
36,000
When p = 500, ℓ = _____
= 72, so 72 laptops would be sold in a month when the store sells
500
36,000
the laptop for $500.
© Houghton Mifflin Harcourt Publishing Company
21. The table gives the number of small figurines that can be placed on a display shelf for various distances
(in centimeters) between them. How many figurines can be placed on the shelf when they are
24 centimeters apart?
Distance between Figurines, s
10
15
20
Number of Figurines, f
12
8
6
120
Because the products sf are 120, f varies inversely as s, and f = ___
s .
120
When s = 24, f = ___
= 5, so 5 figurines can be placed on the shelf when they
24
are 24 centimeters apart.
22. The table gives the amount of water (in gallons) coming out of a garden hose for various amounts of time
(in minutes). How much water comes out of the garden hose in 20 minutes?
Time, t
4
9
16
Amount of Water, w
72
162
288
w
Because the ratios __
t are 18, w varies directly as t, and w = 18t.
When t = 20, w = 18(20) = 360, so 360 gallons of water come out of the garden
hose in 20 minutes.
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Lesson 1
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