2023 YIJC P1 Q11

[It is given that the volume of a right pyramid is \(\frac{1}{3}\times \)base area\(\times \)height.]

A manufacturer designs a decorative item in the shape of a right pyramid with five parts. The square base has sides \(2x\)cm. The four triangular faces each has a base length of \(2x\)cm and height of \(l\) cm. The five parts are joined together as shown in the diagram. The item is made of material of negligible thickness.

The manufacturer determines that the total external surface area of the item must be \(144\)cm\(^{2}\) and that the total volume of the item, \(V\)cm\(^{3}\), should be as large as possible.

1. Show that \(V=8\sqrt{36{{x}^{2}}-2{{x}^{4}}}\).[3]
2. Use differentiation to show that the maximum value of \(V\) is \(72\sqrt{2}\)cm\(^{3}\), obtained when the value of \(x\) is \(3\) cm. Prove that the value of \(V\) is a maximum.[4]

The item is now manufactured with the value of \(x\) being \(3\) cm. To make the item glow in the dark, the item is to be filled entirely with fluorescent liquid. The liquid is injected into the item at a rate of \(\mathrm{10}\)cm\(^{3}\) per second.

3. Find the rate of increase of the depth of the liquid after \(6\) seconds.[5]