9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-14_899_686_212_639}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows the plan of a pool.
The shape of the pool \(A B C D E F A\) consists of a rectangle \(B C E F\) joined to an equilateral triangle \(B F A\) and a semi-circle \(C D E\), as shown in Figure 4.
Given that \(A B = x\) metres, \(E F = y\) metres, and the area of the pool is \(50 \mathrm {~m} ^ { 2 }\),
- show that
$$y = \frac { 50 } { x } - \frac { x } { 8 } ( \pi + 2 \sqrt { } 3 )$$
- Hence show that the perimeter, \(P\) metres, of the pool is given by
$$P = \frac { 100 } { x } + \frac { x } { 4 } ( \pi + 8 - 2 \sqrt { } 3 )$$
- Use calculus to find the minimum value of \(P\), giving your answer to 3 significant figures.
- Justify, by further differentiation, that the value of \(P\) that you have found is a minimum.