Expert: Abe Mantell - 6/5/2009

Question
Hello abe!
My questions are:

1. 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 3 feet per second, how fast is the circumference changing when the radius is 19 feet?

2. A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.4 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 14 cm. (Note the answer is a positive number).

I couldn't figure either of these out on a hw awhile back and we have a final coming up so could you run me through the process with the formulas and the answer so I know how to do these types of examples of related rates. Thanks so much.

Answer
Hello Ray!

1. Since circumference, C=2*pi*r we differentiate w.r.t. "t" to get
.  C'=2*pi*r'  so you see C' is only a function of r' and not r,
.  meaning that it does not matter what he radius is, only how fast it
.  is changing. Using r'=3 ft/sec, we get C'=2*pi*3=6*pi ft/sec

2. Since V=(4/3)*pi*r^3 we differentiate w.r.t. "t" to get
.  V'=4*pi*r^2*r' -- now let r=7 cm (since diam=14) and r'=-0.2 cm/min
.  (since r is half of diam.).  So, V'=4*pi*7^2*(-0.2) cm^3/min
.  Since the question asked at what rate is it *decreasing* we ignore
.  the minus sign to give 39.2pi cm^3/min

OK?

Abe