Question

The radius of a cylinder increases at a rate of $1$ $cm/s$ and its height decreases at a rate of $1$ $cm/s$. Find the rate of change of its volume when the radius is $5cm$ and the height is $15cm$.
If the volume should not change even when the radius and height are changed, what is the relation between the radius and height?

Answer 1

Given $\frac{dr}{dt}=1$ and $\frac{dh}{dt}=-1$

We know that, Volume of a cylinder is given by

$V=\pi r^{2}h$

$⟹\frac{dV}{dt}=\pi 2rh\frac{dr}{dt}+\pi r^2\frac{dh}{dt}$

$⟹\frac{dV}{dt}=\pi r\left(2h\right(1)+r(-1\left)\right)$

$=\pi \times 5(2\times 15-5)$

$=125\pi$