Question

a circular oil slick is expanding at a rate of 2m^2/h. Find the rate at which it's diameter is expanding when it's radius is 1.5m

Answer 1

Answer:
`(4\pi)/(3) \text{ meter per hour}`

Explanation:

The circular oil slick is expanding at a rate of
`2 m^2/h`

Let A be the area of the circular oil slick,

So, the changes in A with respect to time (t),

`(dA)/(dt) = 2`

`(d(\pi r^2))/(dt) = 2`

`2\pi r(dr)/(dt) = 2`

`(dr)/(dt) = (1)/(\pi r)`

Also, the change in diameter with respect to time(t),

`(d)/(dt) (2 r) = 2 (dr)/(dt)`

`(d)/(dt) (2 r) = 2 * (1)/(\pi r)`

`(d)/(dt) (2 r) = (2)/(\pi r)`

For r = 1.5 m,

`(d)/(dt) ( 2 r)]_(r=1.5) = (2)/(\pi * 1.5)=(20)/(\pi * 15)=(4\pi )/(3)\text{ meter per hour}`