Write whether True or False and justify your answer

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1 : 2 : 3.

We have

According to the question, diameter of cone, hemisphere and cylinder are same.

So, radius of cone = radius of hemisphere = radius of cylinder = r

Also, height of cone, hemisphere and cylinder are same.

But in a hemisphere, radius and height always remain same.

So, height of cone = height of hemisphere = height of cylinder = r

Now,

Volume of cone = 1/3 π (radius)² (height)

= 1/3 π (r)² (r) = 1/3 πr³ …(i)

Volume of hemisphere = 2/3 π (radius)³

= 2/3 πr³ …(ii)

Volume of cylinder = π (radius)² (height)

= π r² r = πr³ …(iii)

Now, using equations (i), (ii) and (iii), we can write it in the ratio as

Volume of cone : Volume of hemisphere : Volume of cylinder = 1/3 πr³ : 2/3 πr³ : πr³

= 1/3 : 2/3 : 1

Taking L.C.M of the denominators (3, 3, 1), we get L.C.M as 3. Multiply 3 by each numerator,

Volume of cone : Volume of hemisphere : Volume of cylinder = 3/3 : 6/3 : 3

= 1 : 2 : 3

Hence, it is true.