Question

Two cones of same height 6 cm and radii 3 cm and 4 cm are melted and recast into a cylinder of base radius 5 cm. Find the height of the cylinder.

Answer 1

Given:

Two right circular cones, each of height

h = 6 cm

radii r1 = 3 cm

r2 = 4 cm

They are melted and recast into a cylinder with base radius R = 5 cm.

Find the height H of the cylinder.

= `(1/3) π r^2 h`   ...[Volume of a cone formula]

= π R² H   ...[Volume of a cylinder]

(Use conservation of volume: total volume of the two cones = volume of the cylinder.)

Compute the volume of the two cones.

Volume of cone 1

= `(1/3) π (3)^2 (6)`

= `(1/3) π × 9 × 6`

= 18π cm³

Volume of cone 2

= `(1/3) π (4)^2 (6)`

= `(1/3) π · 16 · 6`

= 32π cm³

Total volume = 18π + 32π

= 50π cm³

Set it equal to the cylinder volume and solve for H.

Cylinder volume

= π (5)² H

= 25π H

25π H = 50π 

⇒ 25H = 50

⇒ H = 2 cm.