P$3\mathrm{.}7\mathrm{.}6**$ Consider the two interacting tank levels shown in Figure P$3\mathrm{.}7\mathrm{.}6$ in which the $F$ s represent volumetric flow rates.

Develop the dynamic models for $H_1$ and $H_2$ assuming that $F_2$ and $F_3$ are given by

$F_2=k_1\sqrt[\phantom{2}]{H_1+H_D}$

$F_3=k_2\sqrt[\phantom{2}]{H_2}$

which represent the effect of the valves in the exit lines and the hydrostatic head for the two tanks. $A_1(15f{t}^2)$ and

$f{t}^2)$are the cross$-$sectional areas of the first and second tanks, respectively. Because the valves have a fixed stem position,

which determines the value of $k,$ the flow dynamics associated with these valves can be assumed instantaneous. Also assume

that the dynamics of the level sensors are instantaneous. The changes in $F_1$ have a time constant ${}_v(3s).$ Initially, this system

is at steady$-$state with $H_1=H_2=5ft$ and $H_D=2ft$ for $F_1=30gpm.$ Find the degrees of freedom. What assumptions are inherent to the modeling? Calculate the final steady$-$state heights before you do any modeling. You can use this to check your python code. Using a Python program to apply your model, submit your graph for the heights as a function of time. determine the two levels in the tanks for

$400$min after a $10\%$ increase in the $(F_1{)}_{\text{s}\text{p}\text{e}\text{c}}$ is implemented. Now determine the two levels in the tanks for $400$ mins after a $10\%$ decrease in F$1$spec. Submit your graph for the heights as a function of time.

Show all steps for all parts and only answer if you will be using python. Please explain where every equation you use comes from and what the coding should look like. Thank you