In the figure below the schematic drawing of a gravity flow tank is given which is in conical form $($Volume $=$ Cross sectional area $x$ height $\frac{?}{3}).$

a$.$ Write the mathematical model of the system for which the probable disturbances are the inlet flowrate, $q_i,$ and the pressure at the exit of discharge pipe, $P_0.$ Exit flow $q_0$ is not a function of height, $h,$ because of the discharge pump, but it is a function of $P_0$ as;

$q_o=\sqrt[\phantom{2}]{\frac{?}{\text{b}\text{a}\text{r}}(p_o)}(\frac{m^3}{s})$ and steady state height in the tank $h_{ss}=1m$;$=1000k\frac{g}{m^3}$

b$.$ Linearize the obtained model and express in terms of deviation variables.