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.

The correct answer is:

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)

The composite function to find the area of the region at time t is.

Given information:

The radius is given by the function . Here, t is time in hours.

As we know that, the area of circular region (circle) is given by the formula .

Now, to get a composite function, replace the value of in area expression, as:

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:

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