A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is

#### Options

1 : 2 : 3

2 : 1 : 3

2 : 3 : 1

3 : 2 : 1

#### Solution

In the given problem, we are given a cone, a hemisphere and a cylinder which stand on equal bases and have equal heights. We need to find the ratio of their volumes.

So,

Let the radius of the cone, cylinder and hemisphere be *x* cm.

Now, the height of the hemisphere is equal to the radius of the hemisphere. So, the height of the cone and the cylinder will also be equal to the radius.

Therefore, the height of the cone, hemisphere and cylinder = *x* cm

Now, the next step is to find the volumes of each of these.

Volume of a cone (*V*_{1}) = `(1/3)pi r^2 h`

`=(1/3)pi (x)^2 (x) `

`=(1/3) pi x^3`

Volume of a hemisphere (*V*_{2})* *= `(2/3) pi r^3`

`=(2/3) pi (x)^3`

`=(2/3) pi x^3`

Volume of a cylinder* *(*V*_{3}) = `pi r^2 h`

`=pi(x)^2(x)`

`=pi x^3`

So, now the ratio of their volumes = (*V*_{1}) : (*V*_{2}) : (*V*_{3})

`=(1/3) pix^3 : (2/3) pi x^3 : pi x^3`

`=(1/3) pi x^3 : (2/3) pi x^3 : (3/3) pi x^3`

= 1: 2 : 3

Therefore, the ratio of the volumes of the given cone, hemisphere and the cylinder is **1: 2:3** .