9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78a994ba-50c5-434f-a060-9596edb505cd-14_554_454_212_744}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height \(h \mathrm {~cm}\). The cross section is a sector of a circle. The sector has radius \(r \mathrm {~cm}\) and angle 1 radian.
The volume of the box is \(300 \mathrm {~cm} ^ { 3 }\).
- Show that the surface area of the box, \(S \mathrm {~cm} ^ { 2 }\), is given by
$$S = r ^ { 2 } + \frac { 1800 } { r }$$
- Use calculus to find the value of \(r\) for which \(S\) is stationary.
- Prove that this value of \(r\) gives a minimum value of \(S\).
- Find, to the nearest \(\mathrm { cm } ^ { 2 }\), this minimum value of \(S\).