PRECALCULUS GT/H
Worksheet 1.10-Direct, Inverse, and Joint Variation
1. If p varies inversely as q , and p  4 when q  9 , find q when p  18 .
2. If y is inversely proportional to the square of x , and y  3 when x  6 , find y when x  3 .
3. If s varies directly as r and inversely as t , and s  10 when r  5 and t  3 , for what value of t
will s  3 when r  4 .
4. If z is jointly proportional to x and y , and z  18 when x  0.4 and y  3 , find z when x  1.2
and y  2 .
5. If r varies directly as p and inversely as the square of q , and r  27 when p  3 and q  2 , find
r when p  2 and q  3 .
6. If z varies jointly as u and v and inversely as w , and z  0.8 when u  8 , v  6 and w  5 , find
z when u  3 , v  10 and w  5 .
7. Kilograms-to-Pounds Problem The number of pounds you weigh is directly proportional to the
number of kilograms you are. Kay Dense steps onto a scale calibrated in kilograms and finds that she
is 50 kilograms. She knows that she weighs 110 pounds.
a) Write the particular equation expressing pounds in terms of kilograms.
b) How many pounds would a person weigh if the scale read:
i) 100 kilograms?
ii) 25 kilograms?
iii) 150 kilograms?
c) How many kilograms would Stan Dupp be if he weighed 165 pounds?
d) What quantity does the proportionality constant represent in the real world?
8. The intensity of light reaching you from a light bulb depends on how far from the bulb you are
standing. Suppose that at 3 meters the intensity is 120 units, and at 6 meters it is 30 units.
a) How does intensity vary with distance? Write the general equation.
b) Find the particular equation.
c) Predict the intensity at 10 meters, at 12 meters.
d) Predict the distance at which the intensity will be 4 units.
e) Draw the graph and tell the real-world significance of the asymptotes.
9. Water Pressure Problem When you swim under water, the pressure in you ears varies directly
with the depth at which you swim. At 10 feet, the pressure is about 4.3 pounds per square inch (psi).
a) Write the particular equation expressing pressure in terms of depth.
b) Predict the pressure at 50 feet.
c) It is unsafe for amateur divers to swim where the pressure is more than 65 psi. How deep can an
amateur diver safely swim?
d) Sketch the graph of pressure versus depth.
10. Wrench Problem The amount of force you must exert on a wrench handle to loosen a rusty bolt
depends on how long the wrench handle is. Suppose that for a particular bolt, a wrench 7 inches long
would require a force of 270 pounds, and a wrench 21 inches long would require 90 pounds.
a) How does force vary with length? Write the general equation.
b) Write the particular equation.
c) What are the units of the proportionality constant? (This quantity is called “torque”, and is a
measure of the amount of “twisting” needed to loosen the bolt.)
d) Find the force needed for wrenches with handles 3 and 30 inches long.
e) How long a wrench would be needed by Sarah Sota, who can exert a force of 300 pounds?
f) How long a wrench would be needed by Sarah’s little sister, Minnie, who can exert a force of 50
pounds?
g) Sketch a graph of force versus handle length in the domain from 3 through 60 inches.
11. Water Main Problem The number of houses that can be served by a water main is directly
proportional to the square of the diameter of the main. This is because the cross-sectional area of the
pipe, and thus the number of liters per minute that can flow, varies directly with the square of the
diameter. Suppose that the City Waterworks has a 10 cm diameter water main that can supply 50
houses.
a) Write the particular equation for this function.
b) How many houses could be served by a water main of diameter 30 cm? 43 cm? 1 meter?
c) A new subdivision of 1500 houses is being planned. What size water main would be needed?
12. Radio Transmitter Problem The strength of a radio signal received from the transmitter varies
inversely with the square of your distance from the transmitter.
a) Derive the particular equation expressing strength in terms of distance if the strength is 1000 units
at a distance of 2 km.
b) Predict the strength 10 kilometers from the transmitter.
c) Predict the strength 100 meters from the transmitter.