8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{18386c8a-6d2d-4c63-972a-bb9f78786b36-30_634_264_319_374}
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\caption{Figure 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{18386c8a-6d2d-4c63-972a-bb9f78786b36-30_762_609_260_1080}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 1 shows the central vertical cross-section, \(O A B C D E O\), of the design for a solid glass ornament.
Figure 2 shows the finite region, \(R\), which is bounded by the \(y\)-axis, the horizontal line \(C B\), the vertical line \(B A\), and the curve \(A O\).
The ornament is formed by rotating the region \(R\) through \(360 ^ { \circ }\) about the \(y\)-axis.
The curve \(A O\) is modelled by the equation
$$x = k y ^ { 2 } + \sqrt { y } \quad 0 \leqslant y \leqslant 4$$
where \(k\) is a constant.
The point \(A\) has coordinates ( \(0.4,4\) ) and the point \(B\) has coordinates ( \(0.4,4.5\) )
The units are centimetres.
- Determine the value of \(k\) according to this model.
- Use algebraic integration to determine the exact volume of glass that would be required to make the ornament, according to the model.
- State a limitation of the model.
When the ornament was manufactured, \(9 \mathrm {~cm} ^ { 3 }\) of glass was required.
- Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.