Question

A cone is placed inside a cylinder as shown. The radius of the cone is half the radius of the cylinder. The height of the cone is equal to the radiusof the cylinder. What is the volume of the cone in terms of the radius, r ?A. V= 1/3pi r^2hB. V= 1/6pi r^2C. V= 1/12pi r^3D. V= 2pi r^3

Answer 1

The radius of the cone is half the radius of the cylinder.

Let r be the radius of cylinder

Then the radius of cone is

`r_c=(r)/(2)`

The height of the cone is equal to the radius of the cylinder

Hence the height of the cone is

`h_c=r`

The formula for volume of a cone is given as:

`V=(1)/(3)\pi r_c^2h_c`

Substitute the radius and height of cone into the formula

`\begin{gathered} V=(1)/(3)\pi((r)/(2))^2r \\ V=(1)/(3)(\pi r^3)/(4) \\ V=(1)/(12)\pi r^3 \end{gathered}`

Therefore the required volume is

`V=(1)/(12)\pi r^3`