A solid sphere of radius 5 cm is melted and recast into a right circular cone with the same base radius (5 cm). Assuming no material is lost, what is the height (in cm) of the cone? Use volume conservation to find the exact height.

Introduction / Context:
This mensuration problem tests volume conservation: when a solid object is melted and recast into another shape without any loss of material, the volumes of the two solids remain equal. Here, a sphere is converted into a right circular cone. Students must know the standard volume formulas: Volume of sphere = (4/3) * π * r^3 Volume of cone = (1/3) * π * R^2 * h The radius of the sphere is given as 5 cm, and the cone has the same base radius, meaning R = 5 cm as well. Because π and the factor 1/3 appear in both formulas, the computation becomes very clean. This is a common aptitude pattern because it tests whether you set up equality correctly and solve for the unknown height h. The main errors come from mixing up radius and diameter, using wrong formulas (like surface area instead of volume), or forgetting that the cone has 1/3 factor. Once the equation is formed, solving for h is straightforward and yields an exact integer.

Given Data / Assumptions:

• Sphere radius r = 5 cm • Cone base radius R = 5 cm (same as sphere radius) • No loss of material ⇒ Volume_sphere = Volume_cone • Required: cone height h

Concept / Approach:
Use volume conservation: (4/3) * π * r^3 = (1/3) * π * R^2 * h. Substitute r = 5 and R = 5. Cancel common factors π and 1/3. Solve the remaining simple equation for h.

Step-by-Step Solution:
1) Write sphere volume: V_sphere = (4/3) * π * r^3 2) Write cone volume: V_cone = (1/3) * π * R^2 * h 3) Set them equal (no loss): (4/3) * π * r^3 = (1/3) * π * R^2 * h 4) Substitute r = 5 and R = 5: (4/3) * π * 5^3 = (1/3) * π * 5^2 * h 5) Cancel (1/3) and π from both sides: 4 * 5^3 = 5^2 * h 6) Compute powers of 5: 5^3 = 125 and 5^2 = 25 7) Solve for h: 4*125 = 25h ⇒ 500 = 25h ⇒ h = 20

Verification / Alternative check:
Compute volumes numerically (without π): Sphere: (4/3)*125 = 500/3. Cone: (1/3)*25*h = 25h/3. Setting equal gives 25h/3 = 500/3 ⇒ h = 20. The same result confirms the algebraic cancellation was correct.

Why Other Options Are Wrong:
• 10 or 15: often come from forgetting the factor 4 in the sphere formula. • 5: happens if you confuse volume with a linear dimension or forget the cone 1/3 factor. • 22: indicates arithmetic error after cancellation.

Common Pitfalls:
• Using diameter (10) instead of radius (5). • Using surface area formulas instead of volume formulas. • Forgetting the cone volume has a 1/3 factor.

Final Answer:
20