$5\mathrm{.}8$ A liquid storage system is shown below. The normal operating conditions are $\frac{?}{\text{b}\text{a}\text{r}}(q{)}_1=10f\frac{t^3}{m}in\frac{,}{b}ar(q{)}_2=5f\frac{t^3}{m}in\frac{,}{b}ar(h)=4ft.$ The tank is $6ft$ in diameter, and the density of each stream is $60l\frac{b}{f}{t}^3.$ Suppose that a pulse change in $q_1$ occurs as shown in Fig. E$5\mathrm{.}8.$

$($a$)$ What is the transfer function relating $H^{\text{'}}$ to $Q_1^{\text{'}}?$

$($b$)$ Derive an expression for $h(t)$ for this input change.

$($c$)$ What is the new steady$-$state value of liquid level $h?$

$($d$)$ Repeat $($b$)$ and $($c$)$ for a change in $q_1$ from $10$ to $20($at $t=0$ to $15f\frac{t^3}{m}in$ at $t=12.$