Write whether the following statements are true or false. Justify your answer : (i) The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere. (ii) The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals the volume of a hemisphere of radius r. (iii) 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.

(i)Let the radius of sphere be r.

Then height of the cylinder, h = 2r

Radius of cylinder = r

Volume of cylinder = r²h

= ×r²×2r

= 2r³

Volume of sphere = (4/3)r³

= (2/3)× 2r³

= (2/3)× Volume of cylinder

Hence the given statement is true.

(ii)Let the edge of the cube is 2r.

So radius of cone = r

Height of cone, h = 2r

Volume of cone = (1/3)r²h

= (1/3)r²×2r

= (2/3)r³

= Volume of a hemisphere of radius r

Hence the given statement is true.

(iii)Let r be radius of cone, hemisphere and cylinder.

So the height of the cone = r

Height of cylinder = r

Volume of cone = (1/3)r²h

= (1/3)r³

Volume of hemisphere = (2/3)r³

Volume of cylinder = r²h

= r³

Ratio of volume of cone , hemisphere and cylinder = (1/3)r³ : (2/3)r³ : r³

= 1/3 : 2/3 : 1

= 1:2:3

Hence the given statement is true.