The ration of cylinder and hemisphere volumes are

Question

the heights of a cone, cylinder and hemisphere are equal. if there radii are in the ratio 2:3:1, then the ratio of their volumes is

Answer 1

The height of a cone , cylinder and hemisphere are equal .

The radius of cone , cylinder and hemisphere are in the ratio 2 : 3 : 1

Cone :

The volume of cone =( 1 / 3 ) *  π r ² h

Let us assume that ( r ) = 2 r  ⇒( 1 / 3 ) * π ( 2 r ) ² h = ( 4 / 3 ) * π  r ² h

Cylinder :

The volume of cylinder  =  π r ² h

Let us assume that ( r ) = 3 r  ⇒ π ( 3 r )  ² h = 9 * π  r ² h

Hemisphere :

The volume of hemisphere. = 2 / 3 π r ³

As height of hemisphere = Radius of hemisphere.

Let us assume that ( r ) = r  ⇒  2 / 3 π r ² * r

The volume of cone , cylinder and hemisphere are in the ratio =

= ( 4 / 3 ) * π  r ² h  : 9 * π  r ² h  : 2 / 3 π r ² * r

So height are equal.

= ( 4 / 3 ) * π  r ²   : 9 * π  r ²   : 2 / 3 π r ²

= ( 4 / 3 ) : 9 :  ( 2 / 3)

= 4 : 27 : 2

The volume of cone , cylinder and hemisphere are in the ratio = 4 : 27 : 2.

Answer 2

The heights of a cone, cylinder and hemisphere are equal.

Therefore .

The radius of cone , cylinder and hemisphere are in the ratio 2 : 3 : 1.

Therefore .

The volumes of cone , cylinder and hemisphere are in the ratio

The height of the sphere is h = d = 2r.

Here radius of the hemisphere to be consider as height, so x = h.

.

Therefore .