7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9963b13-7db4-4a1d-8c75-829f4aade994-12_681_1203_258_376}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C\) with parametric equations
$$x = 3 \tan \theta , \quad y = 4 \cos ^ { 2 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$
The point \(P\) lies on \(C\) and has coordinates ( 3,2 ).
The line \(l\) is the normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
- Find the \(x\) coordinate of the point \(Q\).
The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This shaded region is rotated \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
- Find the exact value of the volume of the solid of revolution, giving your answer in the form \(p \pi + q \pi ^ { 2 }\), where \(p\) and \(q\) are rational numbers to be determined.
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[You may use the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]