N2011 P2 Q2

The diagram shows a rectangular piece of cardboard \(ABCD\) of sides \(n\) metres and \(2n\) metres, where \(n\) is a positive constant. A square of side \(x\) metres is removed from each corner of \(ABCD\). The remaining shape is now folded along \(PQ\), \(QR\), \(RS\) and \(SP\) to form an open rectangular box of height \(x\) metres.

1. Show that the volume \(V\) cubic metres of the box is given by \(V=2n^2x-6nx^2+4x^3\).[3]
2. Without using a calculator, find in surd form the value of \(x\) that gives a stationary value of \(V\), and explain why there is only one answer.[6]