A solid spherical lead ball of radius 6 cm is melted and recast into smaller solid spherical lead balls, each of radius 3 mm. How many such small balls can be formed in total?

Introduction / Context:
This problem is a classic example of volume conservation in solid geometry. A large sphere is melted and recast into several smaller spheres of a different radius. The total volume of material remains the same, so the number of smaller spheres is found by dividing the volume of the large sphere by the volume of one small sphere.

Given Data / Assumptions:

• Radius of the large spherical lead ball R = 6 cm.
• Radius of each small spherical ball r = 3 mm.
• 1 cm = 10 mm, so 3 mm = 0.3 cm.
• Material is conserved: total volume of large sphere = sum of volumes of all small spheres.
• Volume of a sphere with radius r is (4/3)πr³.

Concept / Approach:
Key steps:

• Convert all measurements to the same unit, here centimetres.
• Write the volume of the large sphere using radius 6 cm.
• Write the volume of a small sphere using radius 0.3 cm.
• Divide the larger volume by the smaller volume to get the number of small spheres.
• Cancel common factors and avoid using π explicitly where it cancels.

Step-by-Step Solution:
Step 1: Volume of the large sphere V₁ = (4/3)πR³ = (4/3)π(6)³. Step 2: Compute 6³ = 216, so V₁ = (4/3)π × 216 = 288π cubic centimetres. Step 3: Convert the small radius to centimetres: r = 3 mm = 0.3 cm. Step 4: Volume of a small sphere V₂ = (4/3)πr³ = (4/3)π(0.3)³. Step 5: Compute 0.3³ = 0.027, so V₂ = (4/3)π × 0.027 = 0.036π cubic centimetres. Step 6: Let n be the number of small spheres. Then n × V₂ = V₁, so n = V₁ / V₂ = 288π / 0.036π. Step 7: Cancel π: n = 288 / 0.036. Step 8: Compute the division: 0.036 = 36/1000, so 288 / 0.036 = 288 × (1000/36) = (288/36) × 1000 = 8 × 1000 = 8000.
Verification / Alternative check:
Step 1: Note that the ratio of radii is R / r = 6 / 0.3 = 20. Step 2: The number of small spheres is the cube of the ratio of radii: (R / r)³ = 20³ = 8000. Step 3: This quick method confirms the detailed volume calculation.
Why Other Options Are Wrong:
Option 4000 corresponds to half the correct value and would imply the radius ratio cubed is 10³ instead of 20³. Option 4250 or 8005 are arbitrary and not linked to the exact volume ratio. Option 6000 also does not arise from any correct combination of the radius ratio or volume formulas.
Common Pitfalls:
Using 3 mm directly without converting to centimetres leads to incorrect radius ratio and wrong volume values. Some students mistakenly take the ratio of areas or linear dimensions instead of the ratio of volumes, forgetting that volume scales with the cube of the radius. Another error is to approximate π differently in numerator and denominator instead of recognising that it cancels out completely.
Final Answer:
The total number of small spherical balls that can be formed is 8000.