2025 Nov TZ3 P2 Q8

A manufacturer creates an ice cream model consisting of two parts.
The top part consists of a hemisphere of ice cream with radius \(r\) cm.
The bottom part consists of a cone of ice cream with radius \(r\) cm and height \(h\) cm, which is wrapped in a wafer coating.
This is shown in the following diagram.

1. Consider the case where \(r = 3\) and \(h = 8\).

   [7]

   1. Show that the total volume, \(V\), of ice cream is \(132\) cm\(^3\), correct to \(3\) significant figures.

   2. Determine the curved surface area, \(S\), of the wafer coating.

The manufacturer changes the dimensions of the model to ensure that the total volume, \(V\), of ice cream is \(120\) cm\(^3\).

2. Show that \(h = \frac{360 - 2\pi r^3}{\pi r^2} \).

   [3]

3. Hence, show that the curved surface area of the wafer coating is given by

   \(S = \pi r \sqrt{r^2 + \left( \frac{360 - 2\pi r^3}{\pi r^2} \right)^2}\).

   [2]

The manufacturer wants to use the minimum possible value of \(S\).

4. Determine the minimum value of \(S\) and the corresponding value of \(r\).

   [3]