Consider the liquid$-$level system in Figure P$5\mathrm{.}19,$ where a spherical tank of diameter D is connected to an outlet valve $($R$),$ qin is the input volume flow rate, and pa is the atmospheric pressure. Refer to Problem $5\mathrm{.}15$ for the fluid capacitance of the tank.

$($a$)$ Derive a model of the system with liquid height h as the dynamic variable.

$($b$)$ Obtain a state$-$space model of the system, with qin as the input, and qout as the output. Figure P$5\mathrm{.}19$