A cylindrical tank with a cross sectional area of $5m^2$ is operating at steady state

with an inlet volumetric flowrate of $1\mathrm{.}33\frac{m^3}{m}in.$

There are two outlet streams flowing through the valves and the flowrates are

expressed as follows:

$F_1=\frac{h}{3}$ and $F_2=h^{0\mathrm{.}8}$

where

$F_1$ and $F_2$ : volumetric flowrates of outlet streams, $\frac{m^3}{m}in$

$h$ : level of liquid in the tank, $m$

Assume that densities of all streams are equal.

If the flow to the tank is increased with a step change of $1\frac{m^3}{m}in.$

Answer the following questions:

a$)$ Derive the transfer function between inlet low rate and liquid level and

determine the time constant and static gain.

b$)$ After the change occurs, at what time will the liquid level be $1\mathrm{.}6m?$

c$)$ After the change occurs what is the ultimate value for the liquid level?

d$)$ In another tank with different cross sectional area, at exactly same conditions

the response data is obtained and plotted on the figure. On the same figure

sketch the response profile you obtained. Compare the responses in terms of

static gain and time constant.

Please write your answer by step step and clearly. I'll remember to evaluate your answer.