2025 RI Promo Q12

Fig. 1 shows the net of a right regular hexagon-based prism cut from a rectangular cardboard of one side length 30 cm. The net consists of two regular hexagons of side length \(a\) cm and six rectangles, each with sides of length \(a\) cm and \(h\) cm. The net is folded to form a prism which has a hexagonal base of side length \(a\) cm and vertical height \(h\) cm, as shown in Fig. 2.

1. Show that \(h+a\sqrt{3}=15\).[2]
2. Show that the area of the hexagonal base is \(\dfrac{3\sqrt{3}}{2}a^2\) \(\text{cm}^2\).[1]
3. Use differentiation to find the exact value of the maximum possible volume of the prism and prove that this value is a maximum.[6]
4. Find the ratio of \(h:a\) that gives this maximum volume.[1]
5. Sketch the graph showing the volume of the prism as the value of \(a\) varies.[2]