A food factory wants to set up a new product line. For that purpose, two storage tanks $($diameter of tanks $=4ft)$ are developed by two different companies. Schematic drawing of these tanks are given as below. In system $1,$ the output flow rate changes linearly $($as $q=8\mathrm{.}33h)$ with liquid level. In system $2,$ variation in liquid level doesn't affect output flow rate. $($q; galon$/$min$,$ h;ft$).$ Assume that each system is initially at steady state and the steady state value of $h$ and $q$ is $6ft$ and $50$ gallons$/$min$,$ respectively. For each system calculate the following questions, if the inlet flow rate is suddenly changed from $50$ to $70$ gallons $\frac{?}{\text{m}\text{i}\text{n}}$ at $t=0$;

a$)$ What are the input and output variables for both systems?

b$)$ Determine the controlled and manipulated variables. Indicate the reason for your selection.

c$)$ Derive the differential model of the process.

d$)$ Take the Laplace transform and find the transfer function in terms of deviation variables.

e$)$ By taking the reverse laplace, find how the output variable is changing with respect to time after step input. Draw the graph that shows how the fluid level changes with time.

f$)$ Find the value of the new steady$-$state condition reached as a result of step input.