9758/2020/P2/Q04

Fig. 1 shows the net of a square-based pyramid cut from a square of cardboard of side length $30 \mathrm{~cm}$. The net consists of a square of side length $a \mathrm{~cm}$ and four isosceles triangles, each with base $a \mathrm{~cm}$ and perpendicular height $h \mathrm{~cm}$. The net is folded to form a pyramid which has a square base of side length $a \mathrm{~cm}$ and vertical height $H \mathrm{~cm}$, as shown in Fig. 2.

(i) Show that $H^{2}=225-15 a$.

[2]

(ii) Find the maximum possible volume of the pyramid. You do not need to show that this value is a maximum.

[5] [The volume of a square-based pyramid is $\frac{1}{3} \times$ base area $\times$ height.]

(iii) (a) Find the value of $a$ for which the total surface area of the four triangular faces of the pyramid is a maximum. You do not need to show that this value is a maximum.

[3]

(b) Describe the shape formed from the net in this case.