A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains $\dfrac{1408}{21}$ m$^3$ of air, find the height of the cylindrical part. (Use $\pi = \dfrac{22}{7}$).
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding rag
Model Answer
Let the height of the cylindrical part = $h$ m.
Then, base radius of hemisphere = $r = \dfrac{h}{2}$ m.
Total volume of room = Volume of cylinder + Volume of hemisphere
$$= \pi r^2 h + \frac{2}{3}\pi r^3$$
$$= \pi \left(\frac{h}{2}\right)^2 h + \frac{2}{3}\pi \left(\frac{h}{2}\right)^3$$
$$= \frac{\pi h^3}{4} + \frac{2\pi h^3}{24} = \frac{\pi h^3}{4} + \frac{\pi h^3}{12} = \frac{3\pi h^3 + \pi h^3}{12} = \frac{4\pi h^3}{12} = \frac{\pi h^3}{3}$$
Given volume $= \dfrac{1408}{21}$ m³:
$$\frac{\pi h^3}{3} = \frac{1408}{21}$$
$$\frac{22}{7} \times \frac{h^3}{3} = \frac{1408}{21}$$
$$h^3 = \frac{1408}{21} \times \frac{21}{22} = \frac{1408}{22} = 64$$
$$h = 4 \text{ m}$$
The height of the cylindrical part is 4 m.
Source: Chapter 12, Section 12.3 — Volume of a Combination of Solids
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Explanation
- Key setup: since the base radius of the hemisphere equals the radius of the cylinder, and $r = h/2$, express everything in terms of $h$.
- Volume = $\pi r^2 h + \frac{2}{3}\pi r^3$ (cylinder + hemisphere). Combine carefully to get $\frac{\pi h^3}{3}$.
- Substituting $\pi = \frac{22}{7}$ and the given volume simplifies to $h^3 = 64$, so $h = 4$ m.
- Show all algebraic steps clearly — 3-mark questions reward working, not just the final answer.