Question

I can't figure out how to solve this problem: A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 3 ft/sec. How rapidly is the area enclosed by the ripple increasing at the end of 10 seconds?

Answer 1

start with
\[A=\pi r^2\] take the derivative wrt time, get
\[A'=2\pi r r'\]

Answer 2

you are told \(r'=3\) and \(r=10\) plug in the numbers and you are done

Answer 3

Oh, so it doesn't matter that the question is asking for the rate at a certain time (10 seconds)?

Answer 4

yes it does matter

Answer 5

since \(A'=2\pi r r'\) you have to know what \(r\) is as well as \(r'\)

Answer 6

Wait, how did you know the radius equalled 10?

Answer 7

oh because i am an idiot, and it is not ten at all

Answer 8

if it increases at a rate of \(3\) ft per second, then at the end of 10 seconds it is \(30\) feet

Answer 9

so i was wrong

Answer 10

good catch

Answer 11

Oh wow. Now I feel like the idiot. I should have been able to figure out the radius XD ahaha damn. Thanks for the help!

Answer 12

yw