Question

A cylinder with a radius of 12 cm and a height of 20 cm has the same volume as a cone with a radius of 8 cm. What is the height of the cone

Answer 1

It has a height of 135

Answer 2

Volume of cylinder(V) is given by:

`V = \pi r^2h`

where,

r is the radius and

h is the height of the cylinder.

Volume of cone(V') is given by:

`V'=(1)/(3) \pi r'^2h'`

where, r' is the radius and h' is the height of the cone.

As per the statement:

A cylinder with a radius of 12 cm and a height of 20 cm has the same volume as a cone with a radius of 8 cm

⇒r = 12 cm , h = 20 cm and r' = 8 cm

Since, Volume of cylinder is equal to Volume of cone.

⇒V = V'

then;

`\pi r^2h = (1)/(3) \pi r'^2h'`

⇒
`r^2h = (1)/(3)r'^2h'`

Substitute the given values to solve for h' we have;

`12^2 \cdot 20 = (1)/(3) \cdot 8^2 \cdot h'`

⇒
`144 \cdot 20 = (1)/(3) \cdot 64 \cdot h'`

Multiply both sides by 3 we have;

⇒
`144 \cdot 60=64 \cdot h'`

⇒
`8640=64 \cdot h'`

Divide both sides by 64 we have;

`135 = h'`

or

h' = 135 cm

Therefore, the height of the cone is, 135 cm