# Applications of Composition of Functions in Mathematics

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 cm^{3} of water and is being filled at the constant rate of 100 cm^{3} 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 = π r ^{2}. 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 m ^{2}?
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^{-4}t (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 cm^{3} 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
Composition of Functions |