Question $3($Ordinary Differential Equations$)$

Consider a cylindrical tank of radius $1$ metre and height $3$ metres. Suppose the tank contains a liquid filled to a height of $2$ metres, which has an outlet at the bottom through which the liquid can drain.

If the outlet is opened at time $t=0,$ then the height of the liquid, $h(t),$ decreases over time according to the following differential equation:

$A\frac{(d)h}{(d)t}=-k\sqrt[\phantom{2}]{h}$

where $A=$ is the cross$-$sectional area of the liquid's surface and $k=1$ is a positive constant depending on parameters such as the cross$-$sectional area of the outlet and viscosity of the liquid. Here $h$ is in units of metres $($ m $),A$ is in units of metres squared $(m^2)$ and $t$ is in units of hours $($h$).$

$($a$)$ Solve the differential equation using the separation of variables method, making sure to also apply the initial condition $h(0)=2.$

$($b$)$ What is the height of the liquid at $t=3$ hours?

$($c$)$ After how many hours will the tank be empty?