Answer
The height is increasing at a rate of $~0.38~ft/min$
Work Step by Step
We can write an expression for the radius $r$:
$2r = h$
$r = \frac{h}{2}$
We can differentiate both sides of the equation for volume with respect to $t$:
$V = \frac{1}{3}\pi~r^2~h$
$V = \frac{1}{12}\pi~h^3$
$\frac{dV}{dt} = \frac{1}{4}\pi~h^2~\frac{dh}{dt}$
$\frac{dh}{dt} = \frac{4}{\pi~h^2}~\frac{dV}{dt}$
$\frac{dh}{dt} = [\frac{4}{\pi~(10~ft)^2}~]~(30~ft^3/min)$
$\frac{dh}{dt} = 0.38~ft/min$
The height is increasing at a rate of $~0.38~ft/min$
