a$.$ A cylindrical tank is initially filled with water to a level of ho$.$ The tank has a constant cross section of $A,m^2.$ Water flows out of the tank through a valve at the bottom. The volumetric flow rate through the valve is proportional to the height of water in the tank, $F=\frac{h}{R}$ where $R$ is the resistance with units of $hr\frac{.}{m^2}.$

$($i$)$ Develop an expression for the height of the water in the tank as a function of time, $h(t),$ in terms of the variables $A,R,$ ho$.$

$[10$ marks$]$

$($ii$)$ Calculate the height of water in the tank, if the tank is initially filled to $10m.$

Given: $A=5m^2$ and $R=1hr\frac{.}{m^2}.$

$[10$ marks$]$

b$.$ A wastewater stabilization tank with a cross$-$sectional area $A,m^2$ being filled up with a continuous supply of water flowing at a feed rate of $F_i,\frac{m^3}{s}.$ The liquid level in the tank is to be maintained at a desired height $h,m.$ The effluent from the tank $(F_o,\frac{m^3}{s})$ is controlled by a centrifugal pump. The tank is open from the top. $[$You can answer using the audio record option$]$

Identify the following variables and provide supporting reason for your choice

$($i$.)$ State variables

$[5$ marks$]$

$($ii$.)$ Manipulated Variables

$[5$ marks$]$