College Algebra – 3.4b Inverse and Joint Variation
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Direct Variation – Let x and y denote two quantities. Then y varies directly with x if there is a nonzero constant
k such that
1) Suppose y varies directly with x. If y = 3 when x = 9…
a) Find a general formula relating x and y.
b) What is y when x = 21?
2) The number of gallons of water W used when taking a shower varies directly with the time t in minutes. A
shower lasting 5 minutes uses 30 gallons of water. How much water is used in a shower lasting 11 minutes?
Inverse Variation – Let x and y denote two quantities. Then y varies inversely with x if there is a nonzero
constant k such that
3) Suppose y varies inversely with x. If y = 12 when x = 5…
a) Find a general formula relating x and y.
b) What is y when x = 2?
4) The maximum weight W that can be safely supported by a 2-inch by 4-inch piece of pine varies inversely with
its length l. Experiments indicate that the maximum weight that a 10-foot-long pine 2-by-4 can support is 500
pounds. Find the maximum weight that can be safely supported by a length of 25 feet.
Joint Variation – When a variable quantity Q varies directly with the product of two or more other variables,
we say that Q varies jointly with these quantities.
Combined Variation – Combinations of direct and/or inverse variation may occur. This is usually referred to as
combined variation.
5) Newton’s Formula states that the force of gravitation F between two bodies varies jointly as the product of
their masses m1 and m2 and inversely as the square of the distance between them d. If we use G as the
gravitational constant (of proportionality), write the equation for Newton’s formula.
6) The force F of the wind on a flat surface positioned at a right angle to the direction of the wind varies jointly
with the area A of the surface and the square of the speed v of the wind. A wind of 30 miles per hour blowing on
a window measuring 4 feet by 5 feet has a force of 150 pounds. What is the force on a window measuring 3 feet
by 4 feet caused by a wind of 50 miles per hour?
7) The loss of heat L through a wall varies jointly with the area A of the wall and the temperature difference T
between the inside and outside and varies inversely with the thickness d of the wall. If the temperature
difference is 20° F, a wall with an area of 100 square feet and thickness 0.8 feet has a heat loss of 210 Btu/hr.
Find the heat loss for a wall of the same area having a thickness of 6 inches and a temperature difference of 40° F.