Problem $3.(30$ pts $+5$ pts$)$ Consider the two liquid storage tanks shown in Figure $3.$ The volumetric

flowrate of the stream that connects the two tanks is proportional to the difference between the two levels

$F_1=\frac{h_2-h_1}{R_1},$ which is considered positive when liquid flows from the second to the first tank. The

volumetric flowrate of the exit stream is $F_2=\frac{h_2}{R_2}.$

a$.(15$ pts$)$ Write down a state space model for this system.

b$.(15$ pts$)$ Derive the transfer function between $F_{in}$ and $h_2.$

c$.(+5pts)$ If $A_1=1m^2,A_2=2m^2,A_2=2m^2,R_1=1$min.$m^{-2}$ and $R_2=0\mathrm{.}5$min.$m^{-2},$ calculate

the response of $h_2$ to a step increase in $F_{in}$ by $1\frac{m^3}{m}in.$

Figure $3$