12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-34_396_515_251_772}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The curve shown in Figure 3 has parametric equations
$$x = 6 \sin t \quad y = 5 \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$
The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis.
- Show that the area of \(R\) is given by \(\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \sin t \cos ^ { 2 } t \mathrm {~d} t\)
- Hence show, by algebraic integration, that the area of \(R\) is exactly 20
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-34_451_570_1416_742}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that
- \(x\) and \(y\) are in metres
- the vertical wall of the dam is 4.2 metres high
- there is a horizontal walkway of width \(M N\) along the top of the dam
- calculate the width of the walkway.