A circular oil spill continues to increase in size.
The radius of the oil spill, in miles, is given by the
function , where is the time in hours.
The area of the circular region is given by the
function , where is the radius of the
circle at time .
Explain how to write a composite function to find
the area of the region at time .
DONEIr

Ask by Mathis Erickson. in the United States

Mar 21,2025

Question

A circular oil spill continues to increase in size.
The radius of the oil spill, in miles, is given by the
function , where is the time in hours.
The area of the circular region is given by the
function , where is the radius of the
circle at time .
Explain how to write a composite function to find
the area of the region at time .
DONEIr

Ask by Mathis Erickson. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

We are given the radius as a function of time:

$$
r(t)=0.5+2t
$$

and the area of a circle as a function of its radius:

$$
A(r)=m\,r^2.
$$

To find the area at time $$ t $$, we form a composite function by substituting $$ r(t) $$ into the area function:

$$
A(t)=A(r(t))=m\,[r(t)]^2.
$$

Substitute the expression for $$ r(t) $$:

$$
A(t)=m\,(0.5+2t)^2.
$$

This composite function expresses the area of the oil spill at time $$ t $$.
To find the area of the oil spill at time $$ t $$, substitute the radius function $$ r(t) = 0.5 + 2t $$ into the area function $$ A(r) = m r^2 $$. The composite function is: $$ A(t) = m (0.5 + 2t)^2 $$

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