Question

A cylinder and a cone have the same volume. The cylinder has radius x and height y. The cone has radius 2x. Find the height of the cone in terms of y.

Answer 1

`\large\boxed{height=(3)/(4)y}`

Explanation:

The formulas of a volume of

a cylinder:

`V_1=\pi r^2h`

a cone:

`V_2=(1)/(3)\pi r^2h`

r - radius

h - height

Substitute

The cylinder:

`V_1=\pi x^2y`

The cone:

`V_2=(1)/(3)\pi (2x)^2h=(1)/(3)\pi(4x^2)h=(4)/(3)\pi x^2h`

The cylinder and the cone have the same volume. Therefore we have the equation:

`(4)/(3)\pi x^2h=\pi x^2y` divide both sides by πx²

`(4)/(3)h=y` multiply both sides by 3/4

`(3\!\!\!\!\diagup^1)/(4\!\!\!\!\diagup_1)\cdot(4\!\!\!\!\diagup^1)/(3\!\!\!\!\diagup_1)h=(3)/(4)y\\\\h=(3)/(4)y`