7.
Figure 1
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-06_497_499_397_392}
\captionsetup{labelformat=empty}
\caption{Shape \(X\)}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-06_604_478_349_1069}
\captionsetup{labelformat=empty}
\caption{Shape \(Y\)}
\end{figure}
Figure 1 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 \(Y\).
\section*{8.}
\section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-07_757_1148_354_356}
Figure 2 shows part of the curve with equation
$$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x .$$
The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\). - Show that the equation of the curve may be written as
$$y = x ( x - 3 ) ^ { 2 } ,$$
and hence write down the coordinates of \(A\).
- Find the coordinates of \(B\).
The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
- Find the area of \(R\).