1 Fig. 8 illustrates a hot air balloon on its side. The balloon is modelled by the volume of revolution about the $x$-axis of the curve with parametric equations

$$x = 2 + 2 \sin \theta , \quad y = 2 \cos \theta + \sin 2 \theta , \quad ( 0 \leqslant \theta \leqslant 2 \pi ) .$$

The curve crosses the $x$-axis at the point $\mathrm { A } ( 4,0 )$. B and C are maximum and minimum points on the curve. Units on the axes are metres.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{1601927c-74d7-4cc2-a7f2-2c2a2e8c2c4c-1_812_809_517_704}
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\caption{Fig. 8}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$.
\item Verify that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $\theta = \frac { 1 } { 6 } \pi$, and find the exact coordinates of B .

Hence find the maximum width BC of the balloon.
\item (A) Show that $y = x \cos \theta$.\\
(B) Find $\sin \theta$ in terms of $x$ and show that $\cos ^ { 2 } \theta = x - \frac { 1 } { 4 } x ^ { 2 }$.\\
(C) Hence show that the cartesian equation of the curve is $y ^ { 2 } = x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 }$.
\item Find the volume of the balloon.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4  Q1 [19]}}