Problem $2(30pts).$ A cylindrical tank of radius $R$ is filled with water. The initial height

of the water measured from the base of the tank is $h_0.$ The density of the water is $,$ and

acceleration of gravity is $g.$

In order to empty the tank, at time $t=0$ a hole of radius $b$ is opened at its base. The height

$h(t)$ of the water in the tank is governed by two principles. First, Bernoulli equation $\frac{1}{2}{v}^2=$

$gh(t)$ yields the velocity of the water exiting from the hole. Second, mass conservation

dictates that

$\frac{d}{dt}([$ mass $of],[$ water $in],[$ the tank $])=-([$ mass flux $of],[$ water through $],[$ the hole $])$

$($a$)[7$pts$]$ Using mass conservation and Bernoulli equation derive an ODE for the height of

the water $h(t)$ in the form

$\frac{dh}{dt}=f(h,t)$

$($b$)$ pts