2025 EJC Promo Q7

[It is given that a sphere of radius \(r\) has surface area \(4\pi r^2\) and volume \(\frac{4}{3}\pi r^3\).]

A model of a toy is made up of three parts.

• The top of the toy is modelled by a circular disc of radius \(r\) cm.
• The sides of the toy are modelled by the curved surface of a cylinder of radius \(r\) cm and height \(h\) cm.
• The base of the toy is an empty space which is modelled by the curved surface of a hemisphere of radius \(r\) cm.

The three parts are joined together as shown in the diagram. The model is made of material of negligible thickness.

It is given that the volume of the toy is a fixed value \(k\) cm\(^3\), and the external surface area is a minimum. Use differentiation to find the values of \(r\) and \(h\) in terms of \(k\). Simplify your answers.[7]