Question

An oil slick is expanding as a circle. The radius of the circle is currently 2.5 inches and is increasing at a rate of 6.5 inches per hour. Express the area of the circle, A as a function of h, the number of hours elapsed.

Answer 1

`A = \pi r^2`

So we can express the radius in terms of the hours elapsed like this:

`r = 2.5 + 6.5 h`

And the reason of this is because each hour the radius increase 6.5 inches, and if we replace in the formula of area we got:

`A = \pi (2.5 +6.5 h)^2`

And this function would represent the area of the circle as function of the hours elapsed,
`h\geq 0`

Explanation:

For this case we know the radius of a circle given
`r = 2.5 in` and we also know the incresing rate for the radius:

`(dr)/(dt)= 6.5 (inches)/(\hour)`

And we want to express the are of the circle A as a function of h = the number of hours elapsed.

We know that the area of a circle is given by:

`A = \pi r^2`

So we can express the radius in terms of the hours elapsed like this:

`r = 2.5 + 6.5 h`

And the reason of this is because each hour the radius increase 6.5 inches, and if we replace in the formula of area we got:

`A = \pi (2.5 +6.5 h)^2`

And this function would represent the area of the circle as function of the hours elapsed,
`h\geq 0`