FHSS 2025 P2 Q9

The diagram shows a solid which consists of a cylinder placed on top of a hemisphere. The radius of the cylinder is \(2x\text{ cm}\) and its height is \(y\text{ cm}\). The hemisphere has a radius of \(4x\text{ cm}\). The total volume of the solid is \(200\pi\text{ cm}^3\).

1. Express \(y\) in terms of \(x\), simplifying the expression.[3]
2. Show that the total surface area, \(A\text{ cm}^2\), of the solid is \(A = 40\pi\left(\frac{5}{x} + \frac{2x^2}{15}\right)\).[3]
3. Given that \(x\) can vary, find the stationary value of \(A\).[4]

4. Mr Chan intends to paint the solid with a total surface area found in part (c).

   Determine if the amount of paint needed is a maximum or a minimum.

   [2]