Chapter 3.6, Problem 257E

For the following exercises, use the information in the following table to find $h\text{'}\left(a\right)$ at the given value for a.

$x$	$f\left(x\right)$	$f\text{'}\left(x\right)$	$g\left(x\right)$	$g\text{'}\left(x\right)$
0	2	5	0	2
1	1	-2	3	0
2	4	4	1	-1
3	3	-3	2	3

257. [T] The formula for the volume of a sphere is $S=\frac{4}{3}\pi r^3$ , where r (in feet) is the radius of the sphere. Suppose a spherical snowball in melting in the sun.

a. Suppose $r=\frac{1}{{\left(t+1\right)}^2}-\frac{1}{12}$ where t is time in minutes. Use the chain rule $\frac{dS}{st}=\frac{dS}{dr}\cdot \frac{dr}{dt}$ to find the rate at which the snowball is melting.

b. Use a. to find the rate at which the volume is changing at t = 1 min.