6.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{376d12ab-022c-4070-a1e0-88eacc2fe48e-4_448_803_242_630}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

A river is 30 m wide and flows between two straight parallel banks. At each point of the river, the direction of flow is parallel to the banks. At time $t = 0$, a boat leaves a point $O$ on one bank and moves in a straight line across the river to a point $P$ on the opposite bank. Its path $O P$ is perpendicular to both banks and $O P = 30 \mathrm {~m}$, as shown in Figure 2. The speed of flow of the river, $r \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at a point on $O P$ which is at a distance $x \mathrm {~m}$ from $O$, is modelled as

$$r = \frac { 1 } { 10 } x , \quad 0 \leq x \leq 30$$

The speed of the boat relative to the water is constant at $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t$ seconds the boat is at a distance $x \mathrm {~m}$ from $O$ and is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction $O P$.
\begin{enumerate}[label=(\alph*)]
\item Show that

$$100 v ^ { 2 } = 2500 - x ^ { 2 }$$
\item Hence show that

$$\frac { \mathbf { d } ^ { 2 } x } { \mathbf { d } t ^ { 2 } } + \frac { x } { 100 } = 0$$
\item Find the total time taken for the boat to cross the river from $O$ to $P$.\\
(9)
\end{enumerate}

\hfill \mbox{\textit{Edexcel M4 2008 Q6 [16]}}