A solid right circular cone of height 36 cm and base radius 9 cm is melted to form a solid right circular cylinder of radius 9 cm and height 9 cm. What percentage of the original material is wasted in this process?

Introduction / Context:
This mensuration question tests understanding of volumes of solids and the idea of conservation of material when one solid is melted and recast into another. A cone is melted to form a cylinder with specified dimensions. If the volume of the cylinder is less than that of the original cone, the difference represents material wasted or left over. We need the waste as a percentage of the cone volume, which requires careful use of the cone and cylinder volume formulas.

Given Data / Assumptions:
- The original solid is a right circular cone. - Cone height h_cone = 36 cm, radius r_cone = 9 cm. - The new solid is a right circular cylinder. - Cylinder radius r_cyl = 9 cm, height h_cyl = 9 cm. - Volume of a cone: V_cone = (1 / 3) * π * r^2 * h. - Volume of a cylinder: V_cyl = π * r^2 * h. - We assume uniform density and no other losses besides volume mismatch.

Concept / Approach:
The key idea is to compare volumes. When a cone is melted and used to form a cylinder, the amount of material that actually goes into the cylinder is limited by the cylinder volume. Any extra cone volume is wasted. Therefore, the percentage waste is calculated as (V_cone − V_cyl) / V_cone * 100. Because both shapes use the same radius, many factors simplify when computing the ratio. We only need relative volumes, not the exact numerical value of π, so π can be treated symbolically and cancelled where possible.

Step-by-Step Solution:
Step 1: Write the cone volume formula: V_cone = (1 / 3) * π * (9)^2 * 36. Step 2: Compute 9^2 = 81. So V_cone = (1 / 3) * π * 81 * 36. Step 3: Multiply 81 by 36 to get 2916. Then V_cone = (1 / 3) * 2916 * π = 972π. Step 4: Write the cylinder volume formula: V_cyl = π * (9)^2 * 9 = π * 81 * 9. Step 5: Multiply 81 by 9 to get 729. So V_cyl = 729π. Step 6: Compute the wasted volume: V_waste = V_cone − V_cyl = 972π − 729π = 243π. Step 7: Percentage waste = (V_waste / V_cone) * 100 = (243π / 972π) * 100. Step 8: Simplify the ratio 243 / 972. Divide numerator and denominator by 81 to get 3 / 12. Step 9: Further simplify 3 / 12 = 1 / 4, so percentage waste = (1 / 4) * 100 = 25 percent.

Verification / Alternative check:
Instead of computing the exact volumes, we can directly compare using symbolic expressions. Volume of cone is proportional to h_cone, while volume of cylinder with same radius is proportional to h_cyl. The cone has volume proportional to (1 / 3) * 36 = 12, and the cylinder has volume proportional to 9. So the cone volume to cylinder volume ratio is 12 : 9, which simplifies to 4 : 3. This means the cylinder uses three parts out of four parts of the cone volume, leaving one part wasted. Therefore, 1 out of 4 parts corresponds to 25 percent waste, matching the detailed calculation.

Why Other Options Are Wrong:
Option 5 percent is far too small and would imply almost all of the cone volume is used. Option 0 percent would mean no waste and volumes of the cone and cylinder must be equal, which contradicts the height difference. Option 10 percent underestimates the difference between 972π and 729π and does not match the ratio 4 : 3.

Common Pitfalls:
Learners often forget the (1 / 3) factor in the cone volume formula, which can completely change the ratio. Others incorrectly use the same height for both shapes or misapply the formula for the cylinder. Some students also compute actual numerical values for π unnecessarily, increasing the risk of arithmetic mistakes. It is usually safer to keep π symbolic and cancel it in the ratio step.

Final Answer:
The percentage of material that is wasted in the process is 25%.