2025 SJI PRELIMS P2 Q3

A solid consists of a cuboid \((4-x)\text{ cm}\times(4-x)\text{ cm}\times x\text{ cm}\) with a triangular prism of height \(2x\text{ cm}\) on top.

1. Show that its volume is \(y=2x^3-16x^2+32x\).[3]
2. Explain why \(0<x<4\).[1]
3. Complete the table.[1]

   \(x\)	\(0.5\)	\(1\)	\(1.5\)	\(2\)	\(2.5\)	\(3\)	\(3.5\)
   \(y\)	\(12.25\)	\(18\)	\(18.75\)	\(16\)	\(11.25\)	\(6\)

4. Draw \(y=2x^3-16x^2+32x\) for \(0.5\leq x\leq3.5\).[3]
5. Use the graph to find the smallest \(x\) when the volume is \(15\text{ cm}^3\).[1]
6. Explain how the graph shows that the volume cannot equal \(20\text{ cm}^3\).[1]