A solid is in the shape of a right-circular cone surmounted on a hemisphere, the radius of each of them being 7 cm and the height of the cone is equal to its diameter. Find the volume of the solid.
Generated by claude-sonnet-4-6 · 2026-06-15 10:35 · grounding rag
Model Answer
Given:
- Radius of cone = Radius of hemisphere = r = 7 cm
- Height of cone = diameter = 2r = 14 cm
Volume of the solid = Volume of cone + Volume of hemisphere
$$V = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3$$
$$= \frac{1}{3}\pi r^2(h + 2r)$$
$$= \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times (14 + 2 \times 7)$$
$$= \frac{1}{3} \times \frac{22}{7} \times 49 \times 28$$
$$= \frac{1}{3} \times 22 \times 7 \times 28$$
$$= \frac{4312}{3}$$
$$= \mathbf{1437.33 \ cm^3} \text{ (approx.)}$$
Source: Chapter 12, Section 12.3 – Volume of a Combination of Solids
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Explanation
- The key step is recognising that volume adds directly for combined solids (unlike surface area, where overlapping parts are excluded).
- Height of cone = diameter = 2r = 14 cm — don't use radius here.
- Factor out $\frac{1}{3}\pi r^2$ to simplify calculation.
- Examiners award marks for: correct formula (1 mark), substitution (1 mark), simplification (2 marks), final answer with unit (1 mark).
- Use $\pi = \frac{22}{7}$ unless told otherwise.