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.
a) Show that the volume V(t) of water in the pond at time t satisfies the differential equation dV/dt=k−απ(3a/πh)2/3V2/3,whereαis the coefficient of evaporation.
b) Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable?
c) Find a condition that must be satisfied if the pond is not to overflow.