Section 7.7 Applications Using Rational Equations and Proportions
18. Let x = the maintenance bill on a store that is 4800
square feet.
Bill on shopping center Bill on store
Size of shopping center Size of store
=
19. Let x = the total number of fur seal pups in the
rookery.
Original # tagged # tagged in sample
=
Total # fur seal pups # in sample
20. Let x = the total number of bass in the lake.
Original number Number of tagged
of tagged bass bass in sample
Total number Number of bass
of bass in sample
50 27
=
=
21. Let x= the number of people, in billions, suffering
from malnutrition (in 2010).
28
200 6.9
x
=
9
200 6.9
200 (9)(6.9)
200 62.1
x
x
x
=
=
=
10 inches 23 inches
67 inches
10 23
67
10 (67)(23)
10 1541
154.1
x
x
x
x
x
=
=
=
=
=
Chapter 7 Rational Expressions
25. 18 10
9
x
=
26. 12
18 12
18 12 12
18 144
8
x
x
x
x
=
=⋅
=
=
The length of the side marked x is 8 inches.
28. Notice that the length of the base of the larger
triangle is
()
5x+ inches.
55
x
+
=
29. 20
15 12
15 12 20
15 240
16
x
x
x
x
=
=⋅
=
=
The length of the side marked x is 16 inches.
31. 8
612
x
=
32. Let h = the height of the tree.
86
56
6586
6 430
430 71.7
6
h
h
h
h
=
=⋅
=
=≈
you will go further with the current than against it.
44. makes sense
45. makes sense
It takes the slower skier 1
4 hour longer to complete
the trail than the faster skier, so 1
694
xx
=+.
To solve this equation, multiply both sides by the
LCD, 36.
Section 7.7 Applications Using Rational Equations and Proportions
47. Let x = the usual average rate.
Then 15x− = the average rate, in miles per hour, of the bus in the snowstorm.
Distance
Distance Rate Time Rate
60
Usual conditions 60
xx
=
Multiply both sides of the equation by the LCD,
()
15xx−
() ()
() ()
2
60 60
15 2 15 15
15 60 15 2 60
60 900 2 30 60
xx xx
xx
xxx x
xxxx
−+=−
−
−⋅+ −⋅=
−+ − =
48. Let x= number of hours to fill the pool with all three pipes working.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
First pipe 22
x
x
x
Chapter 7 Rational Expressions
49. Let x = the time, in hours, it takes to prepare one report working together.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
Ben 33
x
x
x
()
50. Let x = the time, in hours, for the experienced carpenter to panel the room.
Then 3x = the time, in hours, for the apprentice to panel the room.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
x
Together, the two carpenters complete one whole job, so 661
3xx
+=
or 621
xx
+=
.
Working alone, it would take the experienced carpenter 8 hours and the apprentice 24 hours to panel the room.
51. Let x=time, in hours, to fill empty swimming pool.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
Normal filling
of pool 22
x
x
x
210
Section 7.7 Applications Using Rational Equations and Proportions
52. Let x= lower interest rate and 1x+=higher interest rate.
Principal Interest rate Interest earned
200
Investment at higher 1 200
interest rate 1
x
x
+
+
53.
()
()()
2
22
25 81 5 9
5959
xx
xx
−= −
=+ −
54.
2
12 36 0
xx
−+=
55. 24
3
yx=− +
slope = 22
33
−
−=
y-intercept = 4
Chapter 7 Rational Expressions
56. a. Substitute to find k.
2
2
2
64 2
ykx
ykx
k
=
=
=⋅
57. a. Substitute to find k.
12 8
12 8
96
k
yx
k
k
k
=
=
=⋅
=
58.
60,000
12,000 40
kA
SP
k
=
⋅
=
7.8 Check Points
1. Since W varies directly with t, we have
.Wkt=
Use the given values to find k.
Wkt
=
2. Beginning with
2
,ykx=
we will use s for the
stopping distance and v for the speed of the car.
Use the given values to find k.
2
2
200 60
200 3600
200
skv
k
k
=
=⋅
=⋅
2
2
18
100
18
100
v
s
s
=
=
Section 7.8 Modeling Using Variation
3. Beginning with
,
k
yx
=
we will use l for the length
of the string and f for the frequency.
Use the given values to find k.
k
fl
=
Find f when
10.l=
5120
5120
10
512
fl
f
f
=
=
=
A string length of 10 inches will vibrate at 512
cycles per second.
4. Let m=the number of minutes needed to solve an
Thus,
8.
x
mp
=
Find m when 8p= and
24.x=
824
8
m
⋅
=
5. Find k:
2
2
120 10 6
120 360
120 360
360 360
Vkhr
k
k
k
π
π
π
=
=⋅ ⋅
=⋅
⋅
=
The volume of a cone having a radius of 12 feet and
a height of 2 feet is
96
π
cubic feet.
7.8 Concept and Vocabulary Check
1.
ykx=
; constant of variation
2.
n
ykx=
k
7.8 Exercise Set
1. Since y varies directly with x, we have
.ykx=
Use the given values to find k.
65 5
ykx
k
=
=⋅
Chapter 7 Rational Expressions
2.
45 5
ykx
k
=
=⋅
3. Since y varies inversely with x, we have
.
k
yx
=
Use the given values to find k.
k
yx
=
4.
63
18
k
yx
k
k
=
=
=
18 2
9
y==
5. Since y varies inversely as x and inversely as the
square of z, we have
2.
kx
yz
=
Use the given values to find k.
2
kx
yz
=
6.
2
74
28
kb
ac
k
k
=
=
=
()
10
25
10 10
5
2
k
k
=
=
The equation becomes
5.
2
yxz=
When x = 8 and z = 12,
()( )
55
812
22
y==
8
()
()
4
12 240.=
8.
()()
()
175 2100 4
175 8400
1
CkAT
k
k
=
=
=
Section 7.8 Modeling Using Variation
9. Since y varies jointly as a and b and inversely as the
square root of c, we have
.
kab
yc
=
Use the given values to find k.
kab
yc
=
10.
2
2
(2)(1)
15 6
kmn
yp
k
=
=
11.
xkyz=
;
Solving for y:
xkyz
=
12.
2
xkyz=
;
Solving for
y
:
2
2
22
xkyz
xkyz
kz kz
=
=
yx
=
14.
3
3
kz
xy
kz
=
Chapter 7 Rational Expressions
15.
kyz
xw
=
;
Solving for y:
kyz
xw
=
16.
2
22
2
2
kyz
xw
wwkyz
x
kz kz w
xw
ykz
=
=
=
18.
()
xkzyw
xkzykzw
xkzw kzy
=−
=−
+=
19.
kz
xyw
=−
;
Solving for y:
kz
xyw
yx
=−
=
20.
()()
kz
xyw
kz
ywx yw
yw
yx xw kz
yx kz xw
=+
+=++
+=
=−
44
0.9
k
=
=
The equation becomes
0.9
TB=
.
When
6
B=
,
0.9(6) 5.4
T==
. The tail length is
Section 7.8 Modeling Using Variation
23. Since B varies directly as D, we have
.
BkD=
Use the given values to find k.
BkD
=
The equation becomes
0.7 .
BD=
When
56
B=
,
56 0.7
D
=
24.
9(12)
9 (12)
dkf
k
k
=
=
=
25. Since a man’s weight varies directly as the cube of
his height, we have
3
.wkh=
Use the given values to find k.
()
()
3
3
170 70
170 343,000
wkh
k
k
=
=
=
26.
2
2
67.5 (45)
dks
k
=
=
22
11 1
(60) (3600) 120
30 30 30
ds== = =
Use the given values to find k.
28 4
k
Br
k
=
=
3.5 feet.
28.
4.4 1000
(1000)4.4 (1000) 1000
k
td
k
k
=
=
=
Chapter 7 Rational Expressions
29. a. Use k
L
R
= to find k.
R
k
LR
=
b. This is an approximate model.
R
30. a. Use k
L
R
= to find k.
k
LR
=
R
=
b. This is an approximate model.
c. 1890
1890 3
634
LR
L
=
=≈
The average life span of a mouse is about 3
years.
31. a. A mammal with a life span of 20 years will have
b. 1890
1890
20
L
R
R
=
=
c. The data for horses is represented on the graph
1890
50
50 1890
1890
50
R
R
R
=
=
=
33. Since intensity varies inversely as the square of the
distance, we have
2
.
k
Id
=
Use the given values to find k.
2
2
.
62.5 3
62.5 9
k
Id
k
k
=
=
=
Section 7.8 Modeling Using Variation
34.
2
2
3.75 40
k
id
k
=
=
35. Since index varies directly as weight and inversely
as the square of one’s height, we have
2
.
kw
Ih
=
Use the given values to find k.
2
kw
Ih
=
The equation becomes
2
703 .
w
Ih
=
When w = 170 and h = 70,
2
703(170) 24.4.
I=≈
36.
(25)
125 20
km
ic
k
=
=
4000
80
4000
80
80 4000
80 4000
c
c
cc c
c
c
=
=⋅
=
10
1200 360
1200 360
360 360
k
k
k
=
=
=
Chapter 7 Rational Expressions
38.
2
2
36 (8)(3)
36 (8)(9)
ekmv
k
k
=
=
=
39. Since intensity varies inversely as the square of the
distance from the sound source, we have
2
k
Id
=.
40.
k
ta
=
41. a. Since the average number of phone calls varies
jointly as the product of the populations and
inversely as the square of the distance, we have
12
2
.
kP P
Cd
=
b. Use the given values to find k.
12
2
kP P
Cd
=
c.
2
0.02(650,000)(490,000)
(400)
39,813
C=
≈
150 18, 000
18,000 150
1
k
k
=
=
43. a.
b. Current varies inversely as resistance. Answers
will vary.
c. Since the current varies inversely as resistance
we have k
RI
=. Use one of the given ordered pairs
Section 7.8 Modeling Using Variation
49. z varies directly as the square root of x and inversely
as the square root of y.
54. does not make sense; Explanations will vary.
Sample explanation: With a positive constant of
variation, both variables will increase.
55. makes sense
56. makes sense
58. Since heat varies directly as the square of the
voltage and inversely as the resistance, we have
2
.
kv
Hr
= To triple the amount of heat generated
For the heat to be tripled, the resistance must be
multiplied by 1
() ()
22
22
2
50 50
7.07 7.07
kk
dd
==
The distance that can be seen is about 7.07 times
farther with the space telescope.
61.
2
32
32
964
3227 0 08
27 18
xx
xxxx
xx
−+
+++−
+
Chapter 7 Rational Expressions
64.
32 3 2
2 5 (5) 2(5) 5
rr−+=−−−+
Chapter 7 Review Exercises
1.
5
624
x
x−
2.
()()
3
25
x
xx
+
−+
Set the denominator equal to 0 and solve for x.
3.
2
2
3
32
x
xx
+
−+
2.x=
4.
2
7
81x+
2
16 4 4 4
xxxx
⋅⋅⋅
8.
()()
()()
2
2
63
318
66
36
3
6
xx
xx
xx
x
x
x
+−
+−=+−
−
−
=−
10.
()
()()
2
2
2
2
22
44
2
yy
yy
yy
yy
y
y
+
+=++
++
=+
12.
()
()
22
22
29
218
23 or 26
xy
xy
xy xy
−
−=
=− + − −
13.
()()
()
222
43 3
12 2 12 2
xx
xx x
xx x x
+−
−⋅= ⋅
++
Chapter 7 Review Exercises
14.
()
()
2
51
553 3
661
5
2
x
xx x
xx
xx
+
+⋅= ⋅
+
+
=
16.
22
2
212 1
55
1
yy yy
y
y
−+ +−
⋅−
−
17.
2
22
2333
4955
yy y
yyy
+− +
⋅
−−
18.
222 4
10 5
xx x+− +
÷
()( )
()
2
25
10 2 4
12 5
10 2 2
1
4
xx
x
xx
x
x
+−
=⋅
+
−+
=⋅
+
−
=
20.
2
17
5
815
y
yy
÷+
++
2
15
7
815
y
yy
+
=⋅
++
()
33
yy
−−
()
2
2
3
42
y
yy
−
+−
=⋅
22.
22
22
88xyxy
xx
+−
÷
8
xy
=−
23.
()
45
420420 4
55 5 5
x
xx
xx x x
+
+
+= = =
++ + +
24.
85418541
3131 31
12 4
31
xxxx
xx x
x
x
−+−++
+=
−− −
−
=−
Chapter 7 Rational Expressions
26.
2
64123
23 23
yy y
yy
−−
−
−−
()
()
()()
2
2
2
64 123
23
64123
23
612
23
2334
23
34
yy y
y
yy y
y
yy
y
yy
y
y
−−−
=−
−−+
=−
−−
=−
−+
=−
=+
28.
()
()
1
55
33 3 13
5
33
525
35
xxx x
xxx x
xx
xx
xxx
xx
−
++
−=−⋅
−− −−−
+
=+
−−
++ +
==
−−
32.
2
32
xx
+
LCD =
2
6x
22
2
7672 53
3323
22
14 15
6
x
xxx
xx
x
x
+=⋅+⋅
+
=
33.
52
1xx
+
+
LCD =
()
1xx+
()
()
()
()
21
52 5
111
52 1522
x
x
x x xx xx
xx xx
+
+= +
+++
++ ++
==
()
2
74
33
xx
+
++
()()
()
74
333
73 4
xxx
x
=+
+++
+
=+
Chapter 7 Review Exercises
35.
2
63
2
4
y
y
y
−+
−
()()
()
2
422
21 2
yyy
yy
−= + −
+= +
LCD =
()()
22yy+−
36.
2
11
1
21
yy
y
yy
−+
−−
−+
()()
11
11 1
yy
yy y
−+
=−
−− −
37.
xy yx
yx
+−
−
LCD =
xy
38.
22
2
21 1
xx
xx x
+
++ −
()()
()()
2
2
21 1 1
111
xx x x
xxx
++=+ +
−= + −
LCD =
()()()
111xxx++−
39.
2
52
11
xx
xx
−
+−
()
()()
22
11 1
111 11
xx
xx xx
+= +
−=− −=−+ −
()
()()
()()
11 11
xx xx
=−
+− +−
()
()()
()()
2
512
552
11 11
xx x xxx
xx xx
−+ −+
==
+− +−
Chapter 7 Rational Expressions
40.
22
44
64xx x
−
−− −
()()
()()
2
2
623
422
xx x x
xxx
−−= + −
−= + −
LCD =
()()()
232xxx+−−
41.
72
3x+
+
LCD =
3x+
42.
2
25 4
69
23
y
yyy
−−
++
()
69323
yy
+= +
2
25 4
69
23
y
yyy
−−
++
()()
()()
()
25 4
32 3 2 3
25 43
y
yyy
yy
−
=−
++
−
x
LCD = x
1
1
xx
x
45.
1
xy
LCD = xy