2025 CJC P2 Q3

Fig. \(1\) shows a net of a hexagonal pyramid folded from a star-shaped cardboard of equal edge length \(a\) cm. The net consists of a hexagon with equal sides of \(x\) cm and six isosceles triangles with base \(x\) cm and side \(a\) cm. The net is folded to form a right pyramid with a hexagonal base of edge length \(x\) cm and vertical height \(h\) cm, as shown in Fig. \(2\). The hexagonal base is made up of six equilateral triangles of side length \(x\) cm.

The volume of a right hexagonal pyramid with base edge \(x\) cm and height \(h\) cm is given by \(V = \frac{\sqrt{3}}{2} x^2 h\)

1. Show that the volume of the hexagonal pyramid, \(V\) satisfies the expression given by \(V^2 = \frac{3}{4}(a^2 x^4 - x^6)\)[2]
2. Find, in terms of \(a\), the maximum possible volume of the hexagonal pyramid. You need not show that this value is a maximum.[4]
3. Find, in terms of \(a\), the total surface area of the hexagonal pyramid when the volume is a maximum.[4]