Question

A circular oil spill continues to increase in size. The radius of the oil spill, in miles, is given by the function r(t) = 0.5 + 2t, where t is the time in hours. The area of the circular region is given by the function A(r) = πr2, where r is the radius of the circle at time t.

Explain how to write a composite function to find the area of the region at time t.

Answer 1

To find the area of the circular oil spill at a given time \( t \), we need to use the concept of composite functions.

1. **Identify the Inner Function**:
The first function we have is the radius function:
\[
r(t) = 0.5 + 2t
\]
This function calculates the radius \( r \) of the oil spill at a specific time \( t \).

2. **Identify the Outer Function**:
The second function is the area function:
\[
A(r) = \pi r^2
\]
This function calculates the area \( A \) of the circular region, given the radius \( r \).

3. **Create the Composite Function**:
To find the area at time \( t \), we want to plug the value of \( r(t) \) into \( A(r) \). This means we will substitute \( r(t) \) into the area function \( A(r) \):
\[
A(t) = A(r(t)) = A(0.5 + 2t)
\]
Now, we substitute \( r(t) \) into the area function:
\[
A(t) = A(0.5 + 2t) = \pi(0.5 + 2t)^2
\]

Thus, the composite function \( A(t) \) represents the area of the region at time \( t \):
\[
A(t) = \pi(0.5 + 2t)^2
\]

This composite function allows you to calculate the area of the oil spill at any specific time \( t \) by simply substituting \( t \) into the function \( A(t) \).