- In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dc3e8e46-c60b-4263-9652-d7c2a322cfae-10_563_561_395_753}
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\caption{Figure 1}
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Figure 1 shows the design for a bathing pool.
The pool, \(P\), shown unshaded in Figure 1, is surrounded by a tiled area, \(T\), shown shaded in Figure 1.
The tiled area is bounded by the edge of the pool and by a circle, \(C\), with radius 6 m .
The centre of the pool and the centre of the circle are the same point.
The edge of the pool is modelled by the curve with polar equation
$$r = 4 - a \sin 3 \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$
where \(a\) is a positive constant.
Given that the shortest distance between the edge of the pool and the circle \(C\) is 0.5 m ,
- determine the value of \(a\).
- Hence, using algebraic integration, determine, according to the model, the exact area of \(T\).