Chapter 1.2, Problem 36P

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

$\frac{dV}{dt}=k-\alpha \pi {\left(\frac{3\alpha }{\pi h}\right)}^{2/3}{V}^{2/3}$,

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.