Volume of a cylinder topped by a cone, then the least curved surface area of the cone for a fixed volume

A glass trophy is made of two solid parts. The lower part is a cylinder of radius \(r\) cm and height \(r\) cm. The upper part is a cone of base radius \(r\) cm and vertical height \(h\) cm, which stands on the top face of the cylinder. The curved surface of the cone is covered with gold leaf.

The diagram below shows the trophy.

1. In this part, \(r=2\) and \(h=6\).
   1. Show that the volume of glass, \(V\text{ cm}^3\), is \(50.3\), correct to \(3\) significant figures.
   2. Find the area of gold leaf, \(S\text{ cm}^2\).[7]

The design is changed so that the volume of glass is \(150\text{ cm}^3\).

2. Show that \(h=\frac{450-3\pi r^3}{\pi r^2}\).[3]
3. Hence, show that the area of gold leaf is given by
   \(S=\pi r\sqrt{r^2+\left(\frac{450-3\pi r^3}{\pi r^2}\right)^2}\).[2]

The designer wants to use as little gold leaf as possible.

4. Find the minimum value of \(S\) and the value of \(r\) for which it occurs.[3]