$3.)$ A river flows into a lake with no outlet. At present, the river has a constant flow rate of

$1,200f\frac{t^3}{s}.$ In the lake, water evaporates from the surface at a rate of $13f\frac{t^3}{s}$ per square

mile of surface area. The surface area of the lake $(A,$ in units of square miles$)$ varies with

depth $(h,$ in units of feet$)$ according to the following equation:

$A=4\mathrm{.}5+5\mathrm{.}5h$

a$)$ Find the equilibrium depth of the lake, such that the river's flow into the lake is

balanced by evaporation from the lake's surface. Hint: If the lake depth is at

equilibrium, think about what that means with respect to mass accumulation in

the lake. $(5$ points$)$

b$)$ Due to increases in upstream extraction, the river's flow rate drops. Below what

flow rate will the lake dry up completely? $(5$ points$)$