Some example word problems, with detailed solutions to explain the possible applications of the Composition of Functions are presented.
Examples with Solutions
Example 1A cylindrical container had 500 cm3 of water and is being filled at the constant rate of 100 cm3 per second. The radius of the container is 50 cm.
a) Write a formula for the quantity Q of water in the container after t seconds.
b) Write a formula for the height H of water in the container in terms of Q.
c) Find an expression for the composition (H o Q)(t) and its meaning.
b) How long does it take the height H of the water in the container to reach 50cm?
Solution to Example 1
Example 2A small stone is thrown into still water and create a circular wave. The radius r of the water wave increases at the rate of 2 cm per second.
a) Find an expression for the radius r in terms of time t (in seconds) after the stone was thrown.
b) If A is the area of the water wave, what is the meaning of the composition (A o r)(t)?
c) Find the area A of the water wave after 60 seconds.
Solution to Example 2
Example 3Starting from 50 meters, the radius r of a circular oil spill increases at the rate of 0.5 meters/second.
a) Express the radius r as a function of time.
b) The area A of a circular shape is given by A = ? r2. Find the composite function (A o r)(t) and explain its meaning.
c) How long will it take the area to be larger 10,000 m2?
Solution to Example 3
Example 4A metallic rod is being heated in a oven where the temperature T varies with the time t as follows: T = 0.2 t + 100 (T in degree Celsius and t in seconds). The length L of the rod varies with temperature and therefore with time according to the formula: L = 100 + 10-4t (L in cm). Find L as a function of the temperature T.
Solution to Example 4
Example 5 ( calculus skills are needed )Air escapes from a balloon at the constant rate of 100 cm3 per second. What is the rate of change of the radius of the balloon (supposed to be a sphere) when r = 10 cm?
Solution to Example 5
More References and linksComposition of Functions
Composition of Functions Questions
Questions on Composite Functions with Solutions.