Answer
a) ${\frac{dV}{dt}}={\frac{1}{3}\pi r^2 \frac{dh}{dt}}$;
b) ${\frac{dV}{dt}}={\frac{2}{3}\pi r h \frac{dr}{dt}}$;
and
c) ${\frac{dV}{dt}}={\frac{1}{3}\pi r^2 \frac{dh}{dt}}+{\frac{2}{3}\pi r h \frac{dr}{dt}}$
Work Step by Step
Volume of a hemisphere $V={\frac{1}{3}\pi r^2 h}$
a) Radius as constant and height is changing, then
${\dfrac{dV}{dt}}={\dfrac{1}{3}\pi r^2 (\dfrac{dh}{dt}})$
b) When height is constant and radius is changing, then
${\dfrac{dV}{dt}}={\dfrac{2}{3}\pi r h (\dfrac{dr}{dt}})$
c) When radius is changing and height is changing, then
${\dfrac{dV}{dt}}={\dfrac{1}{3}\pi r^2 (\dfrac{dh}{dt}})+{\dfrac{2}{3}\pi r h (\dfrac{dr}{dt}})$
