N2013 P2 Q2

Fig. 1

Fig. 2

Fig. 3

Fig. 1 shows a piece of card, \(ABC\), in the form of an equilateral triangle of side \(a\). A kite shape is cut from each corner, to give the shape shown in Fig. 2. The remaining card shown in Fig. 2 is folded along the dotted lines, to form the open triangular prism of height \(x\) shown in Fig. 3.

1. Show that the volume \(V\) of the prism is given by \(V=\frac14x\sqrt3(a-2x\sqrt3)^2\).[3]
2. Use differentiation to find, in terms of \(a\), the maximum value of \(V\), proving that it is a maximum.[6]