Question

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

Answer 1

`A(t) = \pi (0.5 + 2t)^2`

Explanation:

Given

`r(t) = 0.5 + 2t`

`A(r) = \pi r^2`

Required

Determine composite function to find A in terms of t

The interpretation of this question is to determine A(t) and this is solved as follows:

Because:

`r(t) = 0.5 + 2t`

And we want to eliminate r in
`A(r) = \pi r^2`

We have to substitute 0.5 + 2t for r in
`A(r) = \pi r^2`

`A(r) = \pi r^2` becomes

`A(t) = \pi (0.5 + 2t)^2`

The solution can be solved further, but it is best left in this form:

`A(t) = \pi (0.5 + 2t)^2`