2015 AJC P1 Q11

A piece of vanguard sheet, ABCDEF, is in the form of a regular hexagon of side a cm. A kite shape is cut out from each corner to form the shaded shape, as shown in Fig. 1. It is then folded to form the open hexagonal box of height h cm, as shown in Fig. 2.

1. Show that the volume \(V\) of the box in cm3 is given by \(V=2\sqrt{3}\ h{{\left( \frac{\sqrt{3}}{2}a-h \right)}^{2}}\) .[4]
2. Use differentiation to find, in terms of \(a\), the value of h which would result in the volume of the box being maximum.[4]