2026 IGCSE Additional Mathematics May/June 0606/23 Q9

In this question lengths are in centimetres.

The diagram shows a prism whose cross-section is formed by a square of side \(x\) and an equilateral triangle of side \(x\).

The length of the prism is \(y\).

The total surface area of the prism is \(A\).

The volume of the prism is \(250\left(1+\dfrac{\sqrt3}{4}\right)\).

1. Show that \(A=\left(2+\dfrac{\sqrt3}{2}\right)x^2+\dfrac{1250}{x}\).[5]
2. 1. Given that \(x\) varies, find the minimum value of \(A\).[5]
   2. Use \(\dfrac{\mathrm d^2A}{\mathrm dx^2}\) to show that your value is a minimum.[2]