Applications of derevatives

Question

A stone is dropped into a quiet lake and waves move in circles at the speed of 5cm/s .At the instant when radius of circular wave is 8cm , how fast is the enclosed area increasing?

Answer 1

r=8

r' =5

A=pir^2=64pi

A' = 2pirr'

= 2pi8(5)= 80pi cm^2 per sec

when r=8, A=64pi

one second later,

when r=8+5=13 and A=pi(13)^2= 169pi, an increase of 169pi-64pi= 105 pi in one sec

when r=3, A =9pi

one second earlier

r=8-5 =3, A = pi(3)^2 = 9pi, from 9pi to 64pi is an increase of 55pi

(55pi+105pi)/2 = 160pi/2 = 80pi cm^2 per second = rate of change of Area when radius =(13+3)/2 = 8