Answer
a) ${\frac{dV}{dt}}={\pi r^2 \frac{dh}{dt}}$
b) ${\frac{dV}{dt}}={2\pi r h \frac{dr}{dt}}$
c) ${\frac{dV}{dt}}={\pi r^2 \frac{dh}{dt}}+{2\pi r h \frac{dr}{dt}}$
Work Step by Step
The volume of a cylinder is (V) =${\pi r^2 h}$
a) We differentiate both sides, keeping r constant:
${\frac{dV}{dt}}={\pi r^2 \frac{dh}{dt}}$
b) We differentiate both sides, keepking h constant:
${\frac{dV}{dt}}={2\pi r h \frac{dr}{dt}}$
c) We differentiate both sides, treating both r and h as variables:
${\frac{dV}{dt}}={\pi r^2 \frac{dh}{dt}}+{2\pi r h \frac{dr}{dt}}$
