2019 DHS Promo Q12

[A circular cone with base radius \(r\), vertical height \(h\) and slant height \(l\), has curved surfaced area \(\pi rl\) and volume \(\frac{1}{3}\pi r{{h}^{2}}\).]

A capsule made of metal sheet of fixed volume \(p\)cm\(^{3}\) is made up of three parts.

• The top is modelled by the curved surface of a circular cone of radius \(r\)cm. The ratio of its height to its base radius is \(4:3\).

• The body is modelled by the curved surface of a cylinder of radius \(r\)cm and height \(H\)cm.

• The base is modelled by a circular disc of radius \(r\)cm.

The cost of making the body of the capsule is \({\)k}\( per cm\)^{2}$, while that of the top and the base of the capsule is \({\)2k}\( per cm\)^{2}$, where \(k\) is a constant. The total cost of making the capsule is \({\)C}$.

Assume the metal sheet is made of negligible thickness.

1. Show that \(H=\frac{p}{\pi {{r}^{2}}}-\frac{4r}{9}\).[2]
2. Express \(C\) in the form \(\frac{A}{r}+B{{r}^{2}}\), where \(A\) and \(B\) are expressions in terms of \(k\) and \(p\). Use differentiation to show that \(C\) has a minimum as \(r\) varies.[8]
3. Hence determine the ratio of \(H\) to \(r\) when \(C\) is a minimum.[2]