9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d7dcb9f-510c-42c7-bcac-6d6ab3ed6468-28_639_517_255_774}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows the vertical cross-section, \(A O B C D E\), through the centre of a wax candle.
In a model, the candle is formed by rotating the region bounded by the \(y\)-axis, the line \(O B\), the curve \(B C\), and the curve \(C D\) through \(360 ^ { \circ }\) about the \(y\)-axis.
The point \(B\) has coordinates \(( 3,0 )\) and the point \(C\) has coordinates \(( 5,15 )\).
The units are in centimetres.
The curve \(B C\) is represented by the equation
$$y = \frac { \sqrt { 225 x ^ { 2 } - 2025 } } { a } \quad 3 \leqslant x < 5$$
where \(a\) is a constant.
- Determine the value of \(a\) according to this model.
The curve \(C D\) is represented by the equation
$$y = 16 - 0.04 x ^ { 2 } \quad 0 \leqslant x < 5$$
- Using algebraic integration, determine, according to the model, the exact volume of wax that would be required to make the candle.
- State a limitation of the model.
When the candle was manufactured, \(700 \mathrm {~cm} ^ { 3 }\) of wax were required.
- Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.