2025 PHS P2 Q7

The diagram shows an open box with a rectangular base \(x\) cm by \(2x\) cm and four vertical sides of height \(h\) cm. The external surface area of the box is \(36\text{ cm}^2\).

1. Show that the volume, \(V\) cubic centimetres, of the box is \(V=12x-\dfrac23x^3\).[3]
2. Given that \(x\) and \(h\) can vary, find the value of \(x\) for which \(V\) has a stationary value and explain why this value of \(x\) gives the maximum value of volume.[5]