5. In this question you must show all stages of your working. Solutions based entirely on calculator technology are not acceptable.
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\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-10_629_988_370_577}
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\caption{Figure 2}
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A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\), length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Figure 2.
The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)
- Show that the surface area of the brick, \(S \mathrm {~cm} ^ { 2 }\), is given by
$$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
- Hence find the value of \(x\) for which \(S\) is stationary and justify that this value of \(x\) gives the minimum value of \(S\).
- Hence find the minimum surface area of the brick.