Chapter 2.5, Problem 4P

A pond forms as water collects in a conical depression of radius $a$ and depth $h$. Suppose that water flows in at a constant rate $k$ and is lost through evaporation at a rate proportional to the surface area.

Show that the volume $V\left(t\right)$ of water in the pond at time $t$ satisfies the differential equation $dV/dt=k-\alpha \pi {(3a/\pi h)}^{2/3}{V}^{2/3}$ where $\alpha$ is the coefficient of evaporation.

Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable?

Find a condition that must be satisfied if the pond is not to overflow.