A solid cone of height 24 cm and base radius 8 cm is melted and recast into a solid cylinder of radius 6 cm and height 6 cm. What percentage of the original material is wasted in the process?

Introduction / Context:
This is a classic volume comparison problem where one solid is melted and recast into another solid of a different shape. The task is to use volume formulas for a cone and a cylinder, compare the initial and final volumes and interpret the difference as wasted material. Questions like this test both formula recall and percentage calculations.

Given Data / Assumptions:
• Original solid: a right circular cone with height h1 = 24 cm and base radius r1 = 8 cm. • New solid: a cylinder with radius r2 = 6 cm and height h2 = 6 cm. • The cone is melted and recast into the cylinder. • Any material not fitting into the cylinder is considered wasted.

Concept / Approach:
The volume of a cone is V_cone = (1 / 3) * π * r^2 * h. The volume of a cylinder is V_cyl = π * r^2 * h. First compute the volume of the cone and of the cylinder using the given dimensions. The difference V_cone - V_cyl is the volume of material that is not used in the cylinder and is therefore wasted. Finally, compute the percentage waste as (wasted volume / original volume) * 100.

Step-by-Step Solution:
1. Volume of the cone: V_cone = (1 / 3) * π * r1^2 * h1. 2. Substitute r1 = 8 cm and h1 = 24 cm: V_cone = (1 / 3) * π * 8^2 * 24. 3. Compute 8^2 = 64 and 64 * 24 = 1536, so V_cone = (1 / 3) * π * 1536. 4. Simplify: 1536 / 3 = 512, so V_cone = 512π cubic centimetres. 5. Volume of the cylinder: V_cyl = π * r2^2 * h2. 6. Substitute r2 = 6 cm and h2 = 6 cm: V_cyl = π * 6^2 * 6 = π * 36 * 6. 7. Compute 36 * 6 = 216, so V_cyl = 216π cubic centimetres. 8. Wasted volume: V_waste = V_cone - V_cyl = 512π - 216π = 296π. 9. Percentage of material wasted = (V_waste / V_cone) * 100. 10. Substitute: Percentage wasted = (296π / 512π) * 100 = (296 / 512) * 100. 11. Simplify 296 / 512 by dividing numerator and denominator by 8: 296 / 8 = 37, 512 / 8 = 64, so the fraction is 37 / 64. 12. Compute 37 / 64 ≈ 0.578125. Multiply by 100 to get approximately 57.8125 percent. 13. Rounded to one decimal place, this is about 57.8 %.

Verification / Alternative check:
We can also compute the percentage of material used instead of wasted. The used fraction is V_cyl / V_cone = 216π / 512π = 27 / 64, which is about 0.421875 or 42.1875 percent. Subtracting from 100 percent gives the wasted fraction: 100 - 42.1875 = 57.8125 percent. This matches the earlier calculation and confirms that the percentage waste is about 57.8 percent.

Why Other Options Are Wrong:
• 48.5 %: This would correspond to a smaller difference between the cone and cylinder volumes than actually computed. • 37.5 %: This is close to the fraction V_cyl / V_cone, not the wasted portion. • 64 %: This overestimates the waste and would imply the cylinder uses only 36 % of the volume, contrary to the exact ratio 27 / 64. • 52 %: This is an arbitrary mid value and does not match the precise ratio 37 / 64.

Common Pitfalls:
Many students mistakenly equate volumes and forget that part of the melted material does not fit into the new solid. Another common error is to use incorrect dimensions in the formulas or to confuse the cone and cylinder formulas. Additionally, rounding too early can cause small discrepancies; it is best to simplify fractions first and then convert to a percentage.

Final Answer:
The percentage of material wasted is approximately 57.8 %.