8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{545661a6-8d78-488c-b73b-ab2ced60debf-28_663_531_210_258}
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\caption{Figure 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{545661a6-8d78-488c-b73b-ab2ced60debf-28_394_903_431_900}
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\caption{Figure 2}
\end{figure}
Figure 1 shows a sketch of a 16 cm tall vase which has a flat circular base with diameter 8 cm and a circular opening of diameter 8 cm at the top.
A student measures the circular cross-section halfway up the vase to be 8 cm in diameter.
The student models the shape of the vase by rotating a curve, shown in Figure 2, through \(360 ^ { \circ }\) about the \(x\)-axis.
- State the value of \(a\) that should be used when setting up the model.
Two possible equations are suggested for the curve in the model.
$$\begin{array} { l l }
\text { Model A } & y = a - 2 \sin \left( \frac { 45 } { 2 } x \right) ^ { \circ }
\text { Model B } & y = a + \frac { x ( x - 8 ) ( x + 8 ) } { 100 }
\end{array}$$
For each model, - find the distance from the base at which the widest part of the vase occurs,
- find the diameter of the vase at this widest point.
The widest part of the vase has diameter 12 cm and is just over 3 cm from the base.
- Using this information and making your reasoning clear, suggest which model is more appropriate.
- Using algebraic integration, find the volume for the vase predicted by Model B. You must make your method clear.
The student pours water from a full one litre jug into the vase and finds that there is 100 ml left in the jug when the vase is full.
- Comment on the suitability of Model B in light of this information.