Question

The height of a cone is 5 cm. Find the height of another cone whose volume is sixteen times its volume and radius equal to its diameter.

Answer 1

Given:

• Height of first cone h1 = 5 cm.
• Let its radius = r.
• The second cone has a radius equal to the diameter of the first cone, so radius R2 = 2r, and its volume V2 = 16 × V1.

Volume of a cone: V = `(1/3)π r^2 h`

Volume of the first cone: `V1 = (1/3)π r^2 × 5`

Volume of the second cone (radius = 2r, height = h2)

`V2 = (1/3)π (2r)^2 × h2`

= `(1/3)π × 4r^2 × h2`

= `4 × (1/3)π r^2 × h2`

Given V2 = 16 × V1, substitute expressions

`4 × (1/3)π r^2 × h2`

= `16 × (1/3)π r^2 × 5`

Cancel common factor `(1/3)π r^2`

from both sides: 4 h2 = 16 × 5

Solve for h2: h2 = `(16 × 5)/4`

= 4 × 5 

= 20 cm