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.

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)

nishantwork777 nishantwork777 · Answer 2 · 2021-10-25T14:06:15+02:00

The composite function to find the area of the region at time t is[tex]A(r(t))=\pi (0.25+2t+4t^2)[/tex].

Given information:

The radius is given by the function [tex]r(t)=0.5+2t[/tex]. Here, t is time in hours.

As we know that, the area of circular region (circle) is given by the formula [tex]A(r)=\pi r^2[/tex].

Now, to get a composite function, replace the value of [tex]r[/tex] in area expression, as:

[tex]A(r(t))=\pi.(0.5+2t))0.5+2t)\\\\A(r(t))=\pi(0.5\times 0.5+0.5\times 2t\times 2t.0.5+2t\times 2t)\\\\A(r(t))=\pi \times (0.25+2t+4t^2)[/tex]

Therefore, the function can not be simplified any further. Hence, the required function to find the area of the region at time t will be:

[tex]A(r(t))=\pi (0.25+2t+4t^2)[/tex].

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