16.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-44_442_822_285_561}
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\caption{Figure 4}
\end{figure}
Figure 4 shows the design for a container in the shape of a hollow triangular prism.
The container is open at the top, which is labelled \(A B C D\).
The sides of the container, \(A B F E\) and \(D C F E\), are rectangles.
The ends of the container, \(A D E\) and \(B C F\), are congruent right-angled triangles, as shown in Figure 4.
The ends of the container are vertical and the edge \(E F\) is horizontal.
The edges \(A E , D E\) and \(E F\) have lengths \(4 x\) metres, \(3 x\) metres and \(l\) metres respectively.
Given that the container has a capacity of \(0.75 \mathrm {~m} ^ { 3 }\) and is made of material of negligible thickness,
- show that the internal surface area of the container, \(S \mathrm {~m} ^ { 2 }\), is given by
$$S = 12 x ^ { 2 } + \frac { 7 } { 8 x }$$
- Use calculus to find the value of \(x\), for which \(S\) is a minimum.
Give your answer to 3 significant figures.
- Justify that the value of \(x\) found in part (b) gives a minimum value for \(S\).
Using the value of \(x\) found in part (b), find to 2 decimal places,
- the length of the edge \(A D\),
- the length of the edge \(C D\).