2026 GESS P2 Q6

The diagram shows a open-circular cone with radius \(r\) cm and vertical height \(h\) cm. The cone is made from a piece of thin metal sheet and it can contain \(110\pi\) cm\(^3\) of liquid. The cost of the metal sheet is $1.15 per square centimetre. Assume that the thickness of the metal sheet is negligible.

1. Express \(h\) in terms of \(r\).[1]
2. Show that the cost needed to make the cone, $\(C\), is given by \(C=\dfrac{1.15\pi\sqrt{r^6+108900}}r\).[3]
3. Given that \(r\) varies, find the value of \(r\) for which \(C\) has a minimum value.[4]
4. Explain why this value of \(r\) gives the minimum cost needed to make the cone.[1]