This question is Based on Mensuration Chapter of M.L Aggarwal book for ICSE BOARD for class 10.

Write whether the following statements are true or false.

Justify your answer briefly.

This is the Question Number 17, Exercise 17.3 of M.L Aggarwal.

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# 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.

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(i)Let the radius of sphere be r.Then height of the cylinder, h = 2rRadius of cylinder = rVolume of cylinder = r

^{2}h= ×r

^{2}×2r= 2r

^{3}Volume of sphere = (4/3)r^{3}= (2/3)× 2r

^{3}= (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)r

^{2}h= (1/3)r

^{2}×2r= (2/3)r

^{3}= 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)r

^{2}h= (1/3)r

^{3}Volume of hemisphere = (2/3)r

^{3}Volume of cylinder = r

^{2}h= r

^{3}Ratio of volume of cone , hemisphere and cylinder = (1/3)r

^{3}: (2/3)r^{3}: r^{3}= 1/3 : 2/3 : 1

= 1:2:3

Hence the given statement is true.