Question

Water pours out at the rate of $Q$ from a tap, into a cylindrical vessel of radius $r$. Find the rate at which the height of water level rises when the height is $h$.

Answer 1

If $V$ be the volume of liquid in the cylinder, at a height $h$ of the water level then $V=\pi r^{2}h$.
Differentiating both sides w.r.t time $t$, we get
$\frac{dV}{dt}=\pi r^2\frac{dh}{dt}$
$\Rightarrow Q=\pi r^2\frac{dh}{dt}$ or $\frac{dh}{dt}=\frac{Q}{\pi r^2}$
Note the $dV/dt$ represents the rate at which the volume of liquid in the cylinder increases, which is same as the rate of pouring of water through the tap.