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Question

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


Solution


It is given that a cone, a hemisphere and a cylinder stand on equal bases and have the same height.

Let the radius of cone, hemisphere and cylinder be r units.

Radius of the cone = Radius of hemisphere = Radius of cylinder = r

Also,

Height of the cone = Height of the cylinder = Height of the hemisphere

We know that, the height of a hemisphere is same as its radius.

∴ Height of the hemisphere = r

⇒ Height of the cone = Height of the cylinder = Height of the hemisphere = r

Now,

Volume of the cone = 13π × (Radius)2 × Height = 13π × r2 × r = 13πr3

Volume of the hemisphere = 23π × (Radius)3 = 23πr3

Volume of the cylinder = π × (Radius)2 × Height = π × r2 × r = πr3

∴ Volume of the cone : Volume of the hemisphere : Volume of the cylinder

= 13πr3 : 23πr3 : πr3

= 13 : 23 : 1

= 1 : 2 : 3

Thus, the ratio of their volumes is 1 : 2 : 3.

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____.

Mathematics
RD Sharma (2019)
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