Chapter 2.5, Problem 5P

Consider a cylindrical water tank of constant cross section $A$. Water is pumped into the tank at a constant rate $k$ and leaks out through a small hole of area $a$ in the bottom of the tank. From Torricelli’s principle in hydrodynamics (see Problem $6$ in Section $2.3$), it follows that the rate at which water flows through the hole is $\alpha a\sqrt{2gh}$, where $h$ is the current depth of water in the tank, $g$ is the acceleration due to gravity, and $\alpha$ is a contraction coefficient that satisfies $0.5\le \alpha \le 1.0$.

Show that the depth of water in the tank at any time satisfies the equation $dh/dt=(k-\alpha a\sqrt{2gh})/A$.

Determine the equilibrium depth $h_e$ of water, and show that it is asymptotically stable. Observe that $h_e$ does not depend on $A$.