7.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{ffa0b566-6448-491b-96d7-d3806bcfe063-4_556_497_294_342}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffa0b566-6448-491b-96d7-d3806bcfe063-4_549_471_251_1021}
\captionsetup{labelformat=empty}
\caption{Shape \(Y\)}
\end{figure}
Fig. 3 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(A B C D\) joined to a semicircle with \(B C\) as diameter. The length \(A B = d \mathrm {~cm}\) and \(B C = 2 d \mathrm {~cm}\). Shape \(Y\) is a sector \(O P Q\) of a circle with centre \(O\) and radius \(2 d \mathrm {~cm}\). Angle \(P O Q\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,
- prove that \(\theta = 1 + \frac { 1 } { 4 } \pi\).
Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),
- the perimeter of shape \(X\),
- the perimeter of shape \(Y\).
- Hence find the difference, in mm , between the perimeters of shapes \(X\) and \(Y\).