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.

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:

Verification / 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