A cone of radius 8 cm and height 51 cm is melted and made into a cylinder of…

Concept — Melting a solid and recasting it into a new shape neither creates nor destroys material, so the volume of the new solid equals the volume of the original one. A cone of base radius r and height h has volume (1/3)πr2h, while a cylinder of base radius R and height H has volume πR2H. Equating the two volumes leaves a single unknown dimension, and the diameter of a circular base is always twice its radius.

Application

1. Volume of the given cone = (1/3) × π × 82 × 51 cm3.

2. Since 82 = 64 and 51 ÷ 3 = 17, that volume equals π × 64 × 17 = 1088π cm3.

3. Volume of the new cylinder, whose radius R is unknown and whose height is 17 cm, = π × R2 × 17 cm3.

4. Equating the two volumes gives π × R2 × 17 = 1088π.

5. Dividing both sides by 17π gives R2 = 64, so R = 8 cm because a radius is positive.

6. Diameter of the cylinder = 2R = 2 × 8 = 16 cm.

Cross-check — A cone is exactly one-third of the cylinder that stands on the same base with the same height. Here the height of the cone, 51 cm, is three times the height of the cylinder, 17 cm, so tripling the height cancels that one-third factor and the base radius must stay unchanged at 8 cm. Substituting back agrees: π × 82 × 17 = 1088π cm3, the same volume as the cone.

Result — The recast cylinder has radius 8 cm, so its diameter is 16 cm.