9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e48fac26-15a2-4a5e-9204-9d49db8a998a-32_789_452_331_497}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e48fac26-15a2-4a5e-9204-9d49db8a998a-32_681_523_424_1248}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(A B C D E F G H A\) of the bottle with the measurements taken by the student.
The horizontal cross-section between \(C F\) and \(D E\) is a circle of diameter 8 cm and the horizontal cross-section between \(B G\) and \(A H\) is a circle of diameter 2 cm .
The student thinks that the curve \(G F\) could be modelled as a curve with equation
$$y = a x ^ { 2 } + b \quad 1 \leqslant x \leqslant 4$$
where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.
- Find the value of \(a\) and the value of \(b\) according to the model.
- Use the model to find the volume of water that the bottle can contain.
- State a limitation of the model.
The label on the bottle states that the bottle holds approximately \(750 \mathrm {~cm} ^ { 3 }\) of water.
- Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.