(a)[4]
Show that $h$ and $t$ satisfy a differential equation of the form \[ \frac{dh}{dt} = -\frac{B}{2rh^2 - h^2}, \] where $B$ is a positive constant.
(b)[8]
Solve the differential equation and obtain an expression for $t$ in terms of $h$ and $r$.
Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
A tank of water has the form of a hemisphere. Its axis is vertical, the lowest point is $A$ and the radius is $r$, as shown in the diagram. The water depth at time $t$ is $h$. When $t = 0$ the tank is completely full, so the water depth is $r$. At this moment a tap at $A$ is opened and the water starts to leave at a rate proportional to $\sqrt{h}$. The tank is empty at time $t = 14$. When the depth is $h$, the volume of water in the tank is $V$. It is given that $V = \frac{1}{3}\pi(3rh^2 - h^3)$.