The radius of a circular oil slick expands at a rate of 2 m/min.

(a) how fast is the area of the oil slick increasing when the radius is 25 m
(b) if the radius is 0 at time , how fast is the area increasing after 4 mins

We are given:
dr/dt = 2

a)
We know that the area of a circle:
A = πr²
dA/dr = 2πr
We get dA/dt using the chain rule:
dA/dt = dA/dr x dr/dt
= 2πr x 2
= 4πr; r = 25
dA/dt = 4 x 25π
= 100π m²/min

Integrating dr/dt with respect to t,
dr/dt = 2
dr = 2dt
r = 2t + c
r = 0 at t = 0 so c = 0
r = 2t
At t = 4,
r = 8

dA/dt = 4 x 8π
= 32π m²/min