Question

Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of 25π cm²/min. How fast is the radius of the pool increasing when the radius is 6 cm?

Answer 1

`dr/dt = (25)/(12)`cm/min

Explanation:

so from the question, we know that `dA/dt=25pi`, this means that the area of the circular puddle is increasing constantly at this rate.

so in order to find how fast the radius is increasing, we must first determine a relationship between the two values. So, for a circle that is the Area formula `Area = pi*r^2`.

The next part involves related rates and the chain rule.
We know that `dr/dt =dA/dt * dr/dA`
(the rate of radius change expressed as two other rates).
So,
`A=pi*r^2`

`dA/dr = 2pi*r`

`dr/dA = 1/(2pi*r)`

using chain rule now.
`dr/dt = dA/dt *dr/dA`

`dr/dt =25pi * 1/(2pi*r)`

Sub in 6cm for radius
`dr/dt = (25pi)/(2*6pi)`
`dr/dt = (25)/(12)`cm/min