SECTION 7.6 Variation 415
52.The collision impact of an automobile varies jointly as its mass and the square of its
speed. Suppose a 2000-lb car traveling at 55 mph has a collision impact of 6.1. What is
the collision impact of the same car at 65 mph?
53.The weight of a bass varies jointly as its girth and the square
of its length. (Girthis the distance around the body of the fish.)
A prize-winning bass weighed in at 22.7 lb and measured 36 in.
long with a 21-in. girth. How much would a bass 28 in. long
with an 18-in. girth weigh?
54.The weight of a trout varies jointly as its length and the square
of its girth. One angler caught a trout that weighed 10.5 lb and
measured 26 in. long with an 18-in. girth. Find the weight of a
trout that is 22 in. long with a 15-in. girth.
55.The force needed to keep a car from skidding on a curve varies inversely as the radius of
the curve and jointly as the weight of the car and the square of the speed. If 242 lb of force
keeps a 2000-lb car from skidding on a curve of radius 500 ft at 30 mph, what force would
keep the same car from skidding on a curve of radius 750 ft at 50 mph?
56.The maximum load that a cylindrical column
with a circular cross section can hold varies
directly as the fourth power of the diameter of the
cross section and inversely as the square of the
height. A 9-m column 1 m in diameter will
support 8 metric tons. How many metric tons can
be supported by a column 12 m high and m in
diameter?
57.The number of long-distance phone calls between two cities during a certain period varies
jointly as the populations of the cities, and , and inversely as the distance between
them. If 80,000 calls are made between two cities 400 mi apart, with populations of
70,000 and 100,000, how many calls are made between cities with populations of 50,000
and 75,000 that are 250 mi apart?
58.In 2007, 51.2% of the homes in the United States used natural gas as the primary heating
fuel. (Source:U.S. Census Bureau.) The volume of gas varies inversely as the pressure and
directly as the temperature. (Temperature must be measured in Kelvin(K), a unit of measure-
ment used in physics.) If a certain gas occupies a volume of 1.3 L at 300 K and a pressure of
18 newtons, find the volume at 340 K and a pressure of 24 newtons.
59.A body mass index from 27 through 29 carries a slight risk of weight-related health prob-
lems, while one of 30 or more indicates a great increase in risk. Use your own height and
weight and the information in Example 7to determine your BMI and whether you are at risk.
60.The maximum load of a horizontal beam that is supported at both ends varies directly as
the width and the square of the height and inversely as the length between the supports.
A beam 6 m long, 0.1 m wide, and 0.06 m high supports a load of 360 kg. What is the
maximum load supported by a beam 16 m long, 0.2 m wide, and 0.08 m high?
p 1 p 22
39 mLoad = 8 metric tons1 mFind each square root. If it is not a real number, say so. See Section 1.3.
66.Does the relation define yas a function of x? Give its domain using interval
notation. See Section 3.5.
y=x- 1