Question

A rock is thrown into a still pond and causes
a circular ripple. If the radius of the ripple is
increasing at a rate of 4 ft/sec, at what speed
is the area of the ripple increasing when its
radius is 6 feet?

Answer 1

oh my favorite ridiculous word problem! i guess we have to do it.

Answer 2

you know that
\[A=\pi r^2\] and you are looking for
\[\frac{dA}{dt}\] right? the change is A with respect to time.

Answer 3

taking the derivative of
\[A=\pi r^2\] we get
\[A'(t)=2\pi r r'(t)\] by the chain rule. you are told that
\[f'(t)=4\] so this means
\[A'(t)=8\pi r\]

Answer 4

sorry that should read
\[r'(t)=4\]

Answer 5

now replace r by 6 and you get your answer

Answer 6

OHHHHH! Gotchaaaaaa