A large cylindrical tank contains a hole of area A at its bottom....

Let $h$ be the height of water in the tank at any time $t$.

The rate of volume of water entering the tank is given by $Q_{in} = A v$.

The velocity of efflux from the hole at the bottom of the tank is $v_{eff} = \sqrt{2gh}$.

The rate of volume of water leaving the tank is $Q_{out} = A \sqrt{2gh}$.

The water level will rise as long as the rate of inflow is greater than the rate of outflow, i.e., $Q_{in} > Q_{out}$.

The water level will stop rising when the rate of inflow becomes equal to the rate of outflow.

$A v = A \sqrt{2gh}$

$v = \sqrt{2gh}$

$h = \dfrac{v^2}{2g}$

Thus, the water level will rise to a height $\dfrac{v^2}{2g}$ and then stop rising.