Answer
(a) $11.2cm/min$
(b) $14.9cm/min$
Work Step by Step
Identify the given quantities: $\frac{dV}{dt}=10m^3/min$, $\frac{h}{d}=\frac{3}{8}$, , $h=4m$. Thus the base radius $r=d/2=\frac{8(4)}{2(3)}=\frac{16}{3}$, $r=4h/3$
(a) $V=\pi r^2h/3=16\pi h^3/27$, differentiate $\frac{dV}{dt}=\frac{16}{9}\pi h^2\frac{dh}{dt}=\frac{16}{9}\pi 4^2\frac{dh}{dt}=\frac{256}{9}\pi \frac{dh}{dt}$. As $\frac{dV}{dt}=10m^3/min$, we have $\frac{dh}{dt}\approx0.112m/min=11.2cm/min$
(b) With $r=4h/3$, we have $\frac{dr}{dt}=\frac{4}{3}\frac{dh}{dt}=14.9cm/min$
