2015 PJC P1 Q8

Fig. \(1\) shows a card in the form of a square of fixed side \(a\). A triangle is cut from each side, to give the shape shown in Fig. \(2\). The remaining card shown in Fig. \(2\) is folded along the dotted lines, to form the right pyramid with square base of side \(x\) as shown in Fig. \(3\).

1. Show that the volume \(V\) of the pyramid is given by \(V=\frac{{{x}^{2}}}{3}\sqrt{\frac{{{a}^{2}}-\sqrt{2}ax}{2}}\).[4]
2. Use differentiation to find in surd form the value of \(\frac{a}{x}\) that gives a stationary value of \(V\).[4]