Expert: Ahmed Salami - 1/5/2010

Question

hi. good day. i got this question from TC7.

A stone is dropped into a still pond, and concentric circular ripples spread out. Find the rate of change of the disturbed region when its radius is (a) 4cm, and (b) 7cm.

I was able to answer this but I'm not sure of my answer. My answers are 8 and 14, respectively. Am I right?
thank you. i would really appreciate your help.


Answer
Hi Jessie,
The disturbed region is simply the area enclosed by the circle whose radius is given. At any given radius r, the disturbed region has an area
A = πr²
Differentiating with respect to t,
dA/dt = π.d(r²)/dt
     = π.(2r)dr/dt
     = 2πr(dr/dt)
This means that the rate of change of the disturbed region dA/dt is always 2πr times bigger than the rate of change of the radius dr/dt.
At r = 4
dA/dt = 8π(dr/dt)
At r = 7
dA/dt = 14π(dr/dt)
Notice that you need the rate of change of the radius to find the rate of change of the disturbed region. It would probably be given.

Hope it helps you.

Regards