Applications of Differentiation

Fig. 1 shows a piece of card in the form of a square of side \({2\sqrt{2}\ a}\). Four isosceles triangles of height \({\sqrt{2}\ x}\) are cut from the card to give the shape shown in Fig. 2. The remaining card shown in Fig. 2 is folded along the dotted lines, to form the square pyramid as shown in Fig. 3. If \({V}\) is the volume of the square pyramid, show that \({V = k(a - x)^2\ \sqrt{ax}}\), where \({k}\) is an exact real constant to be determined. Hence, using differentiation, find the maximum value of \({V}\) in terms of \({a}\).[9]

[Volume of a right square pyramid is \(\frac{1}{3}\)(base area)(perpendicular height).]