MathematicsMensuration

A solid sphere of radius 6 cm is melted and recast into smaller spheres of radius 2 cm. How many such spheres can be formed?

Correct Answer

📐 MA • Math Concept & PrincipleMathematicsMensuration

Option B

Quick Summary:

When a solid object is melted and recast into smaller objects of the same material, the total volume remains conserved. By equating the volume of the large solid sphere to the combined volume of $n$ smaller spheres, we find that the number of spheres formed is 27. Thus, Option B is the correct choice.

🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Imagine melting a large ball of clay into several small, equal-sized marbles; the total amount of clay stays the same, allowing you to count how many marbles you can make by dividing the total volume.

🧠 Memory Mnemonic / Shortcut Aid

Volume stays true, divide old by new!

🔢 Key Formulas

$V = \frac{4}{3} \pi r^3$ — Volume of a sphere of radius $r$

$n = \frac{V_{large}}{V_{small}} = \left(\frac{R}{r}\right)^3$ — Number of recast spheres

⚙️ Working Principle

The principle of conservation of volume dictates that the total volume of matter remains constant during phase changes or physical reshaping when no material is lost. The volume of a sphere is given by the formula $V = \frac{4}{3} \pi r^3$ . Dividing the volume of the larger sphere by the volume of one smaller sphere yields the total number of smaller spheres that can be produced.

📌 Key Points

▸
Volume conservation is the fundamental principle behind melting and recasting problems.
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The radius of the large sphere is $R = 6\text{ cm}$ and the radius of the small sphere is $r = 2\text{ cm}$ .
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The ratio of the radii is $\frac{6}{2} = 3$ , and cubing this ratio gives $3^3 = 27$ spheres.

✅ Advantages

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Allows efficient determination of material distribution during manufacturing and shaping processes.

❌ Disadvantages / Limitations

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Assumes zero material loss or wastage during melting and recasting.

🛠️ Applications / Uses

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Foundry work and metal casting
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Glass blowing and reshaping plastic components

📄 Additional Information

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Constant: $\pi \approx \frac{22}{7}$ or $3.14159$
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Option A (18), Option C (36), and Option D (54) are incorrect because they fail to correctly apply the cube of the radius ratio.

📊 Diagram / Illustration

✅

B is correct — 27 smaller spheres of radius 2 cm can be formed from a solid sphere of radius 6 cm based on volume conservation.

Core Concepts Used

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Conservation of Volume Sphere Mensuration Recasting ratio

💡 EXAM TIP

For similar mensuration problems involving melting and recasting, always express the ratio of volumes in terms of the ratio of linear dimensions raised to the power of three for 3D shapes.

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