A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1:2:3.

Explanation

A cone, a hemisphere and a cylinder stand on equal bases and same height.
Let radius of each of a cone, a hemisphere and cylinder be r (equal bases)
Heightof hemisphere = r,
So, the height of the cone = r,
and height of cylinder = r.
Now, volume of cone, `V_(1)=(1)/(3)pi r^(2)h=(1)/(3)pi r^(2)xx r=(1)/(3)pi r^(3)`
Volume of hemisphere, `V_(2)=(2)/(3)pi r^(3)`
and volume of cylinde, `V_(3)=pi r^(2)h=pi r^(2)xx r=(3)/(3)pi r^(3)`
`therefore V_(1):V_(2):V_(3)=(1)/(3)pi r^(3):(2)/(3)pi r^(3) : (3)/(3)pi r^(3)=1 : 2 : 3`
i.e.,the ratio of their volumes is 1 : 2 : 3.