Q1. [2] § 12.2 Surface Area of a Combination of Solids § 12.3 Volume of a Combination of Solids § 12.4 Summary
A solid piece of metal in the form of a cuboid of dimensions $11\,\text{cm} \times 7\,\text{cm} \times 7\,\text{cm}$ is melted to form '$n$' number of solid spheres of radii $\frac{7}{2}$ cm each. Find the value of $n$.
Previously asked in CBSE board exam
2022 30/2/1 Q3
Generated by claude-sonnet-4-6 · 2026-06-15 10:36 · grounding rag
Model Answer
Volume of cuboid = $11 \times 7 \times 7 = 539 \text{ cm}^3$
Volume of one sphere $= \dfrac{4}{3}\pi r^3 = \dfrac{4}{3} \times \dfrac{22}{7} \times \dfrac{7}{2} \times \dfrac{7}{2} \times \dfrac{7}{2} = \dfrac{4}{3} \times \dfrac{22}{7} \times \dfrac{343}{8} = \dfrac{539}{3} \text{ cm}^3$
Since metal is melted and recast:
$$n = \frac{\text{Volume of cuboid}}{\text{Volume of one sphere}} = \frac{539}{\dfrac{539}{3}} = 3$$
$$\boxed{n = 3}$$
Source: Chapter 12, Section 12.3 (Volume of a Combination of Solids)
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Explanation
- The key principle: when a solid is melted and recast, total volume is conserved.
- Equate volume of cuboid = $n \times$ volume of one sphere, then solve for $n$.
- Use $\pi = \frac{22}{7}$ and $r = \frac{7}{2}$ cm; the numbers cancel cleanly to give $n = 3$.
- Show both volume calculations clearly for full marks (1 mark each step typically).
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