Answer
a) 3053.63 $in^{3}$, 48858.049 $in^{3}$
b) $\frac{d}{dr}$$\frac{4}{3}$ $\pi$$r^{3}$ = 4$\pi$$r^{2}$ $\frac{dr}{dt}$
Work Step by Step
a)
r = 9 in
$\frac{dr}{dt}$ = 3 in/min
V = $\frac{4}{3}$ $\pi$$r^{3}$
$\frac{d}{dr}$ V = $\frac{d}{dr}$$\frac{4}{3}$ $\pi$$r^{3}$
= 4$\pi$$r^{2}$ $\frac{dr}{dt}$
=4$\pi$ $(9^{2})$ (3)
= 3053.63 $in^{3}$
r = 36 in
$\frac{dr}{dt}$ = 3 in/min
= 4$\pi$$r^{2}$ $\frac{dr}{dt}$
=4$\pi$ $(36^{2})$ (3)
=48858.049 $in^{3}$
b)
$\frac{dr}{dt}$ is constant but $\frac{dV}{dt}$ is not because when you find the derivative of the volume, the volume is not constant.
r increases when $\frac{dV}{dt}$ increases
$\frac{d}{dr}$$\frac{4}{3}$ $\pi$$r^{3}$ = 4$\pi$$r^{2}$ $\frac{dr}{dt}$
