4.05: Surface-area optimisation
A prism has cross-section made from a square of side \(x\) and an equilateral triangle of side \(x\), and length \(y\). Its volume is \(250(1+\sqrt3/4)\).
(a)[5]
Show that its total surface area is \(A=\left(2+\dfrac{\sqrt3}{2}\right)x^2+\dfrac{1250}{x}\).
See the official mark scheme below.
(b) (i)[5]
Find the minimum value of \(A\).
See the official mark scheme below.
(b) (ii)[2]
Use a second derivative to show that this value is a minimum.
See the official mark scheme below.
