Question

Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?

Answer 1

The area of the pool increasing at the rate of 125.6 when the radius is 5 cm

Explanation:

Given:

radius of the pool increases at a rate of 4 cm/min

To Find:

How fast is the area of the pool increasing when the radius is 5 cm?

Solution:

we are given with the circular pool

hence the area of the circular pool =

A =
`\pi r^2`-----------------------------(1)

The area of the pool is increasing at the rate of 4 cm/min, meaning that the area of the pool is changing with respect to time t

so differentiating eq (1) with respect to t , we have

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

we have to find
`(dA)/(dt)` with
`(dr)/(dt)` = 4 cm/min and r = 5 cm

substituting the values

`(dA)/(dt) =\pi *2(5)*4`

`(dA)/(dt) =\pi * 10*4`

`(dA)/(dt) =\pi * 40`

`(dA)/(dt) =40\pi`

`(dA)/(dt) =125.6`