1. Given that \(( x - 2 )\) is a factor of \(2 x ^ { 3 } + k x - 4\), find the value of the constant \(k\).
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2. (a)
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The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(A B C D E F\). The base \(A C F D\) of the roof is a horizontal rectangle, and the cross-section \(A B C\) of the roof is an isosceles triangle with \(A B = B C\). The lengths of \(A C\) and \(C F\) are \(2 x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively, and the height of \(B E\) above the base of the roof is \(x \mathrm {~cm}\). The total surface area of the five faces of the roof is \(600 \mathrm {~cm} ^ { 2 }\) and the volume of the roof is \(V \mathrm {~cm} ^ { 3 }\). Show that \(V = k x \left( 300 - x ^ { 2 } \right)\), where \(k = \sqrt { a } + b\) and \(\alpha\) and \(b\) are integers to be determined.
(b) Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum.
(c) Find the maximum volume of the roof. Give your answer in \(\mathrm { cm } ^ { 3 }\), correct to the nearest integer.
(d) Explain why, for this roof, \(x\) must be less than a certain value, which you should state.
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