A right circular cone (solid metal) of height 8 cm and base radius 2 cm is melted and recast into a sphere. Find the radius of the sphere.
Aptitude
Volume and Surface Area
Difficulty: Easy
Choose an option
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A2 cm
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B3 cm
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C4 cm
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D5 cm
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ENone of these
Answer
Correct Answer: 2 cm
Explanation
Introduction / Context:This is a volume-conservation recasting problem: a cone is melted and remade into a sphere. Since material is conserved, cone volume equals sphere volume; we solve for the new radius.
Given Data / Assumptions:
- Cone height h = 8 cm.
- Cone radius r = 2 cm.
- No loss of material; V_cone = V_sphere.
Concept / Approach:
- V_cone = (1/3)*π*r^2*h.
- V_sphere = (4/3)*π*R^3.
- Equate and solve for R.
Step-by-Step Solution:
V_cone = (1/3)*π*(2)^2*(8) = (1/3)*π*32 = (32/3)π cm^3Set (32/3)π = (4/3)π*R^3 ⇒ cancel (π, 1/3): 32 = 4*R^3R^3 = 8 ⇒ R = 2 cmVerification / Alternative check:Back-substitute: V_sphere = (4/3)π*8 = (32/3)π, exactly the cone volume; consistent.
Why Other Options Are Wrong:
- 3, 4, 5 cm: Give sphere volumes that are too large.
- None of these: Not applicable; 2 cm is exact.
Common Pitfalls:
- Using diameter in place of radius in volume formulas.
- Arithmetic errors when equating and simplifying.
Final Answer:2 cm