\includegraphics{figure_9}

Fig. 9 shows a hemispherical bowl, of radius 10 cm, filled with water to a depth of $x$ cm. It can be shown that the volume of water, $V$ cm$^3$, is given by

$$V = \pi(10x^2 - \frac{1}{3}x^3).$$

Water is poured into a leaking hemispherical bowl of radius 10 cm. Initially, the bowl is empty. After $t$ seconds, the volume of water is changing at a rate, in cm$^3$ s$^{-1}$, given by the equation

$$\frac{dV}{dt} = k(20 - x),$$

where $k$ is a constant.

\begin{enumerate}[label=(\roman*)]
\item Find $\frac{dV}{dx}$, and hence show that $\pi x \frac{dx}{dt} = k$. [4]

\item Solve this differential equation, and hence show that the bowl fills completely after $T$ seconds, where $T = \frac{50\pi}{k}$. [5]
\end{enumerate}

Once the bowl is full, the supply of water to the bowl is switched off, and water then leaks out at a rate of $k$ cm$^3$ s$^{-1}$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Show that, $t$ seconds later, $\pi(20 - x) \frac{dx}{dt} = -k$. [3]

\item Solve this differential equation.

Hence show that the bowl empties in $3T$ seconds. [6]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4 2012 Q9 [18]}}