10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-15_538_529_205_744}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a solid brick in the shape of a cuboid measuring \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The total surface area of the brick is \(600 \mathrm {~cm} ^ { 2 }\).
- Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the brick is given by
$$V = 200 x - \frac { 4 x ^ { 3 } } { 3 }$$
Given that \(x\) can vary,
- use calculus to find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
- Justify that the value of \(V\) you have found is a maximum.