A student was asked to make a model shaped like a cylinder with two cones attached to its ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its total length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model.
Generated by claude-sonnet-4-6 · 2026-06-15 10:34 · grounding rag
Model Answer
Given: Diameter = 3 cm → radius (r) = 1.5 cm; Total length = 12 cm; Height of each cone = 2 cm.
Height of cylinder = 12 − 2 − 2 = 8 cm
The model consists of 1 cylinder + 2 cones.
Volume of cylinder:
$$V_{\text{cyl}} = \pi r^2 h = \frac{22}{7} \times (1.5)^2 \times 8 = \frac{22}{7} \times 2.25 \times 8 = \frac{396}{7} = 56.57 \text{ cm}^3$$
Volume of 2 cones:
$$V_{2\text{ cones}} = 2 \times \frac{1}{3}\pi r^2 h = \frac{2}{3} \times \frac{22}{7} \times (1.5)^2 \times 2 = \frac{2}{3} \times \frac{22}{7} \times 4.5 = \frac{66}{7} = 9.43 \text{ cm}^3$$
Total volume of air in the model:
$$V = 56.57 + 9.43 = \boxed{66 \text{ cm}^3}$$
Source: Surface Areas and Volumes, Section 12.3 (Exercise 12.2, Q.2)
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Explanation
- The key step is finding the cylinder's height by subtracting both cone heights from total length: 12 − 2 − 2 = 8 cm.
- Volume of the whole model = Volume of cylinder + Volume of two cones (not one). Students often forget to double the cone volume.
- The answer simplifies neatly to exactly 66 cm³ — examiners expect this clean value.
- Use $\pi = \dfrac{22}{7}$ as instructed in Exercise 12.2.