Fig. 8.1 shows an upright cylindrical barrel containing water. The water is leaking out of a hole in the side of the barrel.

\includegraphics{figure_8.1}

The height of the water surface above the hole $t$ seconds after opening the hole is $h$ metres, where
$$\frac{dh}{dt} = -A\sqrt{h}$$
and where $A$ is a positive constant. Initially the water surface is 1 metre above the hole.

\begin{enumerate}[label=(\roman*)]
\item Verify that the solution to this differential equation is
$$h = \left(1 - \frac{1}{2}At\right)^2.$$ [3]
\end{enumerate}

The water stops leaking when $h = 0$. This occurs after 20 seconds.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the value of $A$, and the time when the height of the water surface above the hole is 0.5 m. [4]
\end{enumerate}

Fig. 8.2 shows a similar situation with a different barrel; $h$ is in metres.

\includegraphics{figure_8.2}

For this barrel,
$$\frac{dh}{dt} = -B\frac{\sqrt{h}}{(1+h)^2},$$
where $B$ is a positive constant. When $t = 0$, $h = 1$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Solve this differential equation, and hence show that
$$h^{\frac{1}{2}}(30 + 20h + 6h^2) = 56 - 15Bt.$$ [7]
\item Given that $h = 0$ when $t = 20$, find $B$.

Find also the time when the height of the water surface above the hole is 0.5 m. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4 2014 Q8 [18]}}