A conical tank is being filled, from an initially empty state, with a constant volumetric inflow qin of liquid. The radius of the tank varies linearly with the depth (h) such that r = a h. The bottom of the tank is the apex of the cone. Due to corrosion there is a small hole with area A0 near the bottom of the tank; liquid leaks through the small hole. The goal of this problem is to find the time-varying height of liquid in the tank, h(t).

a. Suppose that the rate of water leaking through the hole is described by qleak =A0*sqrt(2gh(t)) with g equal to the gravitational constant. find the1st Order (nonlinear) ODE for h(t).

b. Derive an equation to predict the steady-state liquid level in the tank, .

c. Define ?(?) = sqrt((?)/ss), separate variables and derive an implicit solution for x(t) and determine an equation for the constant ?:

? = 1/? (ln(1 ?) + ? ^5/5 + ? ^4/ 4 + ? ^3/ 3 + ? ^2/2 + ?)

d. Sketch h(t) and compare to the case with no leaks.