15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-44_647_917_260_484}
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\caption{Figure 4}
\end{figure}
Figure 4 shows a solid wooden block. The block is a right prism with length \(h \mathrm {~cm}\). The cross-section of the block is a semi-circle with radius \(r \mathrm {~cm}\).
The total surface area of the block, including the curved surface, the two semi-circular ends and the rectangular base, is \(200 \mathrm {~cm} ^ { 2 }\)
- Show that the volume \(V \mathrm {~cm} ^ { 3 }\) of the block is given by
$$V = \frac { \pi r \left( 200 - \pi r ^ { 2 } \right) } { 4 + 2 \pi }$$
- Use calculus to find the maximum value of \(V\). Give your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
- Justify, by further differentiation, that the value of \(V\) that you have found is a maximum.