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

we know that

The radius of the oil spill, in miles, is given by the function

$r(t) = 0.5 + 2t$--------> $equation 1$

The area of the circular region is given by the function

$A(r) = \pi *r^{2}$--------> $equation 2$

Substitute equation $1$ in equation $2$

$A(t) = \pi *(0.5+2t)^{2}$

$A(t) = \pi *(0.5+2t)^{2}\\ A(t)=\pi *(0.25+2t+4t^{2} )\\ A(t)=\pi *(4t^{2} +2t+0.25)$

therefore

the answer is

A composite function to find the area of the region at time t is equal to

$A(t)=\pi *(4t^{2} +2t+0.25)$

Answer 2

The correct answer is:

$A(r(t))=\pi(0.25+2t+4t^2)$

Explanation:

To write a composite function, we apply one function to another given function. In this case, we want to find area in terms of time; this means that the function A(r) gets applied to the function r(t).

In order to do this, we replace r with r(t). We already know that r(t)=0.5+2t; this means we replace r with 0.5+2t:

A(r(t))=π(0.5+2t)²

To simplify this, we simplify the squared term:

A(r(t)) = π(0.5+2t)(0.5+2t)

A(r(t)) = π(0.5*0.5+0.5*2t+2t*0.5+2t*2t)

A(r(t)) = π(0.25+t+t+4t²)

A(r(t)) = π(0.25+2t+4t²)