8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{725966d1-d29d-4c9d-b850-c67d55cdd6e8-11_691_1098_365_513}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a solid cuboid \(A B C D E F G H\).
\(A B = x \mathrm {~cm} , B C = 2 x \mathrm {~cm} , A E = h \mathrm {~cm}\)
The total surface area of the cuboid is \(180 \mathrm {~cm} ^ { 2 }\).
The volume of the cuboid is \(V \mathrm {~cm} ^ { 3 }\).
a. Show that \(V = 60 x - \frac { 4 x ^ { 3 } } { 3 }\)
Given that \(x\) can vary,
b. use calculus to find, to 3 significant figures, the value of \(x\) for which \(V\) is a maximum. Justify that this value of \(x\) gives a maximum value of \(V\).
c. Find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).