Problem $1.$ A tall cylindrical tank is being filled, from an initially empty state $)=0$;$h=(0,$ by a constant inflow IN$,$ Q $($L$/$s$),$ of liquid. The tank bottom is flat and has corroded and is leaking through a small hole of area, $A_0.$ If the cross$-$sectional area of the tank is denoted by $A,$ and the time$-$varying fill height of the liquid is $h(t),$ then complete the following below. Tip: formulate the problem by defining a general mass balance using flow rate IN$,$ flow OUT and relate this to the accumulation of tank volume, $V=Ah(t).$

a$.$ Find the differential equation $($dh$/$dt$)$ describing the tank height dynamics if the volumetric leak rate OUT $(Q_0)$ obeys the following functional form:

$Q_o=A_o\sqrt[\phantom{2}]{2gh(t)},$ where $g$ is the gravitational acceleration.

b$.$ Determine the final steady$-$state liquid height $(h_{ss})$ in the tank Accumulation $=0.$

c$.$ Define $x=\sqrt[\phantom{2}]{h},$ separate variables and deduce the implicit solution for $h.$

d$.$ Sketch out the tank height $(h)$ versus time $(t)$ and compare this with the case for a nonleaking tank.