14.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-23_650_1182_212_383}
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\caption{Figure 6}
\end{figure}
Figure 6 shows a solid triangular prism \(A B C D E F\) in which \(A B = 2 x \mathrm {~cm}\) and \(C D = l \mathrm {~cm}\).
The cross section \(A B C\) is an equilateral triangle.
The rectangle \(B C D F\) is horizontal and the triangles \(A B C\) and \(D E F\) are vertical.
The total surface area of the prism is \(S \mathrm {~cm} ^ { 2 }\) and the volume of the prism is \(V \mathrm {~cm} ^ { 3 }\).
- Show that \(S = 2 x ^ { 2 } \sqrt { 3 } + 6 x l\)
Given that \(S = 960\),
- show that \(V = 160 x \sqrt { 3 } - x ^ { 3 }\)
- Use calculus to find the maximum value of \(V\), giving your answer to the nearest integer.
- Justify that the value of \(V\) found in part (c) is a maximum.
\includegraphics[max width=\textwidth, alt={}, center]{b85872d4-00b2-499b-9765-f7559d3de66a-24_63_52_2690_1886}