3.6 Variation 373
5. .? = fcv\ where ^ is the strength o f a muscle that has length x
2
mv
6. / =
, where / i s the centripetal force o f an object o f mass m moving along a
circle o f radius r at velocity v
Concept
k>0.
Check
Match each statement with its corresponding
7. v varies directly as x.
8. >' varies inversely as x.
( v = kx)
2
y
B.
2
C.
y
[y ™= ^)
10. x varies directly as the second power
ofy.
(x - ky )
9. v varies directly as the second power
o f x. (y = kx )
A.
graph. In each case,
Solve each variation problem. See Examples
D. y
y
I-4.
11. I f v varies directly as .v, and y = 10 when .v = 2, findj' when x = —6.
12.
I f y varies directly as x, and y = 3 when x — 10, find y when x = 40.
13.
I f m varies jointly as x and y, and /w = 10 when x = 4 and y = 7, find m when
x = 11 and v = 8.
14.
I f m varies jointly as z and
and p = 1'.
15.
I f y varies inversely as x, and >' = 10 when x " 3, find 3' when jc = 12.
and w = 10 when z = 3 and p = 5, find m when r = 5
16. If_y varies inversely as x, and>» = 20 when x = 5, find;' when x = 20.
17.
Suppose r varies directly as the square o f m, and inversely as s. I f r =* 12 when
w
6 and ,y = 4, find r when m = 4 and s = 10.
18.
Suppose p varies directly as the square o f z, and inversely as r. I f p *= y when z = 4
and r = 10. find /? when r = 2 and r = 16.
2
3
19. Let a be directly proportional to m and « , and inversely proportional to >' . I f a — 9
when m =• 4, n = 9, and v = 3, find a when ;w = 6, « = 2, and >' = 5.
20.
2
2
I f y varies directly as x, and inversely as m and r , and v = f when x = 1, m = 2,
and ;• = 3, find j> when x = 3, m = 1, and r = 8.
5o/ve rac/; problem. See Examples
1-4.
21. Circumference
of a Circle
The circumference o f a circle varies directly as the
radius. A circle with radius 7 in. has circumference 43.96 in. Find the circumference
o f the circle i f the radius changes to 11 in.
22.
Pressure Exerted by a Liquid
The pressure exerted by a certain liquid at a given
point varies directly as the depth o f the point beneath the surface o f the liquid. The
pressure at 10 ft is 50 pounds per square inch (psi). What is the pressure at 15 ft?
23.
Resistance of a Hire
The resistance in ohms o f a platinum wire temperature sensor varies directly as the temperature in degrees Kelvin ( K ) . I f the resistance is
646 ohms at a temperature o f 190 K , find the resistance at a temperature o f 250 K.
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
374
CHAPTER 3 Polynomial and Rational Functions
24.
Distance to the Horizon
The distance that a person can see to the horizon on a clear
day from a point above the surface o f Earth varies directly as the square root o f the
height at that point. I f a person 144 m above the surface o f Earth can see 18 km to the
horizon, how far can a person see to the horizon from a point 64 m above the surface?
25.
Weight on the Moon
The weight o f an object on Earth is directly proportional to
the weight o f that same object on the moon. A 200-lb astronaut would weigh 32 lb
on the moon. How much would a 50-lb dog weigh on the moon?
26.
Water Emptied by a Pipe The amount o f water emptied by a pipe varies directly
as the square o f the diameter o f the pipe. For a certain constant water flow, a pipe
emptying into a canal w i l l allow 200 gal o f water to escape in an hour. The diameter o f the pipe is 6 in. How much water would a 12-in. pipe empty into the canal in
an hour, assuming the same water flow?
27.
Hooke's Law for a Spring
Hooke's law for an elastic
spring states that the distance a spring stretches varies
directly as the force applied. I f a force o f 15 lb stretches
a certain spring 8 in., how much w i l l a force o f 30 lb
stretch the spring?
:
«gg
-r""~
"^ZZ
;jp
8 in.
i
15 lb
28.
Current in a Circuit
The current in a simple electrical circuit varies inversely as
the resistance. I f the current is 50 amps when the resistance is 10 ohms, find the current i f the resistance is 5 ohms.
29. Speed of a Pulley The speed o f a pulley varies inversely as its diameter. One kind
of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed o f
a similar pulley with diameter 5 in.
30.
Weight of an Object The weight o f an object varies inversely as the square o f its
distance from the center o f Earth. I f an object 8000 mi from the center o f Earth
weighs 90 lb, find its weight when it is 12,000 m i from the center o f Earth.
3 1 . Current Flow
In electric current flow, it is found that the resistance (measured in
units called ohms) offered by a fixed length o f wire o f a given material varies
inversely as the square o f the diameter o f the wire. I f a wire .01 in. in diameter has
a resistance o f .4 ohm, what is the resistance o f a wire o f the same length and material with diameter .03 in. to the nearest ten-thousandth?
32.
Illumination
The illumination produced by a light source varies inversely as the
square o f the distance from the source. The illumination o f a light source at 5 m is
70 candela. What is the illumination 12 m from the source?
33. Simple Interest
Simple interest varies jointly as principal and time. I f S1000 left
at interest for 2 yr earned S110, find the amount o f interest earned by S5000 for 5 yr.
34.
Volume of a Gas Natural gas provides 35.8% o f U.S. energy. (Source: U.S.
Energy Department.) The volume o f a gas varies inversely as the pressure and d i rectly as the temperature in degrees Kelvin (K). I f a certain gas occupies a volume
of 1.3 L at 300 K. and a pressure o f 18 newtons per square centimeter, find the
volume at 340 K and a pressure o f 24 newtons per square centimeter.
|BI Force of Wind The force o f the wind blowing on a vertical surface varies jointly
as the area o f the surface and the square o f the velocity. I f a wind o f 40 mph exerts
a force o f 50 lb on a surface o f j f t , how much force w i l l a wind o f 80 mph place
on a surface o f 2 ft ?
2
2
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
3.6 Variation 375
36.
" \ 1
Exercises;
Volume of a Cylinder
The volume o f a right
circular cylinder is jointly proportional to the
square o f the radius o f the circular base and to
the height. I f the volume is 300 c m when the
height is 10.62 cm and the radius is 3 cm, find
the volume to the nearest tenth o f a cylinder
with radius 4 cm and height 15.92 cm.
3
10.6:
V= 300 c m
37.
Sports Arena Construction
The roof o f a
new sports arena rests on round concrete
pillars. The maximum load a cylindrical
column o f circular cross section can hold
varies directly as the fourth power o f the
diameter and inversely as the square o f the
height. The arena has 9-m tall columns that
are 1 m in diameter and w i l l support a load
of 8 metric tons. How many metric tons w i l l
be supported by a column 12 m high and j m
in diameter?
3
9m
Load = 8 metric tons
38. Sports Arena Construction
The sports arena in Exercise 37 requires a beam 16 m
long, 24 cm wide, and 8 cm high. The maximum load o f a horizontal beam that is
supported at both ends varies directly as the width and square o f the height and
inversely as the length between supports. I f a beam o f the same material 8 m long,
12 cm wide, and 15 cm high can support a maximum o f 400 kg, what is the maximum load the beam in the arena w i l l support?
39.
Period of a Pendulum
The period o f a pendulum varies directly as the square root
o f the length o f the pendulum and inversely as the square root o f the acceleration due
to gravity. Find the period when the length is 121 cm and the acceleration due to
gravity is 980 cm per second squared, i f the period is 6 ir seconds when the length is
289 cm and the acceleration due to gravity is 980 cm per second squared.
40.
Long-Distance Phone Calls
The number o f long-distance phone calls between
two cities in a certain time period varies directly as the populations p and p o f the
cities, and inversely as the distance between them. I f 10,000 calls are made between
two cities 500 m i apart, having populations o f 50,000 and 125.000, find the number
of calls between two cities 800 mi apart, having populations o f 20,000 and 80,000.
t
2
4 1 . Body Mass Index
The federal government has developed the body mass index
( B M I ) to determine ideal weights. A person's B M I is directly proportional to his or
her weight in pounds and inversely proportional to the square o f his or her height in
inches. (A B M I o f 19 to 25 corresponds to a healthy weight.) A 6-foot-tall person
weighing 177 lb has B M I 24. Find the B M I (to the nearest whole number) o f a person whose weight is 130 lb and whose height is 66 in. (Source: Washington Post.)
42.
Poiseuille's Law According to Poiseuille's law. the resistance to flow o f a blood
vessel, R, is directly proportional to the length, /, and inversely proportional to
the fourth power o f the radius, r. I f R = 25 when / = 12 and r = .2, find R to the
nearest hundredth as r increases to .3, while / is unchanged.
43. - Stefan-Boltztnann
Law The Stefan-Boltzmann law says that the radiation o f heat
R from an object is directly proportional to the fourth power o f the Kelvin temperature o f the object. For a certain object, R = 213.73 at room temperature (293 K ) .
Find R to the nearest hundredth i f the temperature increases to 335 K.
Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
376 CHAPTER 3 Polynomial and Rational Functions
44.
Nuclear Bomb Detonation
Suppose a nuclear bomb is detonated at a certain site.
The effects o f the bomb w i l l be felt over a distance from the point o f detonation that
is directly proportional to the cube root o f the yield o f the bomb. Suppose a
100-kiloton bomb has certain effects to a radius o f 3 k m from the point o f detonation. Find the distance to the nearest tenth that the effects would be felt for a 1500kiloton bomb.
45.
Malnutrition
Measure
A measure o f malnutrition, called the pelidisi,
varies
directly as the cube root o f a person's weight in grams and inversely as the person's
sitting height in centimeters. A person with a pelidisi below 100 is considered to be
undernourished, while a pelidisi greater than 100 indicates overfeeding. A person
who weighs 48,820 g with a sitting height o f 78.7 cm has a pelidisi o f 100. Find the
pelidisi (to the nearest whole number) o f a person whose weight is 54,430 g and
whose sitting height is 88.9 cm. Is this individual undernourished or overfed?
Weight: 48,820 g
46.
Photography
Weight: 54,430 g
Variation occurs in a formula from photography. In the formula
25F
L =
2
,
st
the luminance, L , varies directly as the square o f the F-stop, F, and inversely as the
product o f the film ASA number, s, and the shutter speed, /.
(a)
What would an appropriate F-stop be for 200 A S A film and a shutter speed o f
255 sec when 500 footcandles o f light are available?
(b) I f 125 footcandles o f light are available and an F-stop o f 2 is used with 200 A S A
film, what shutter speed should be used?
Concept Check
47.
Work each problem.
For k > 0, i f y varies directly as x, then when x increases, v
, and
when x decreases, v
48.
For k > 0, i f y varies inversely as x, then when x increases, y
, and
when x decreases, y
49.
What happens to y i f y varies inversely as x, and x is doubled?
50.
I f v varies directly as x, and x is halved, how is v changed?
51. Suppose y is directly proportional t o x , andx is replaced by \x. What happens to v?
52.
53.
What happens to y i f y is inversely proportional to x, and x is tripled?
Suppose p varies directly as r and inversely as r . I f ;• is halved and t is doubled,
what happens to pi
54.
I f m varies directly as p and a , and p doubles while q triples, what happens to ml
3
1
4
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