Question

A solid right circular cylinder of radius 8 cm and height 2 cm is melted and cast into a right circular cone of height 3 times that of the cylinder. Find the curved surface of the cone.

Answer 1

Given:

Cylinder radius = 8 cm, height = 2 cm.

Cone height = 3 × (cylinder height) = 6 cm. (Use volume formula for solids and curved-surface formula for a cone.)

Volume of the cylinder = π r² h

= π × 8² × 2

= 128π cm³

Let R be the radius of the cone. Volume of the cone

= `(1/3)π R^2 H`

= `(1/3)π R^2 × 6`

= 2π R². Equate volumes (cylinder → cone): 2π R^2 = 128π ⇒ R^2 = 64 ⇒ R = 8 cm. (Method of equating volumes when metals are recast.)

Slant height of the cone l = `sqrt(R^2 + H^2)`

= `sqrt(8^2 + 6^2)`

= `sqrt100`

= 10 cm

Curved surface area of the cone

= π R l = π × 8 × 10

= 80π cm²

Curved surface of the cone

= 80π cm² ≈ 251.33 cm²