N2010 P1 Q9

A company requires a box made of cardboard of negligible thickness to hold \(300\) cm\(^{3}\) of powder when full. The length of the box is \(3x\) cm, the width is \(x\) cm and the height is \(y\) cm. The lid has depth \(ky\) cm, where \(0 < k\le 1\) (see diagram).

1. Use differentiation to find, in terms of \(k\), the value of \(x\) which gives a minimum total external surface area of the box and the lid.[6]
2. Find also the ratio of the height to the width, \(\frac{y}{x}\), in this case, simplifying your answer.[2]
3. Find the values between which \(\frac{y}{x}\) must lie.[2]
4. Find the value of \(k\) for which the box has square ends.[2]