(Q5)A solid consisting of a right circular cone standing on a hemisphere, is placed upright in a right circular cylinder full of water and touching the bottom. Find the volume of water left in the cylinder, given that the radius of the cylinder is 3 cm and its height is 6 cm. The radius of the hemisphere is 2 cm and the height of the cone is 4 cm.

DOCLMANB222433

In the figure drawn here,

ABCD is a cylinder, LMN is a hemisphere and OLM is a cone, We know that when a solid is immersed in the water, then water displaced equal to the volume of the solid.

Volume of the Cylinder = πr2h = π × ( )2 × = π cm3

Volume of the hemisphere =
2
3
 πr3 =
2
3
 π × ( )3 =
3
 π cm3
Volume of the cone =
1
3
 πr2h =
1
3
 × π × ( )2 × =
3
 πcm3
Volume of the solid figure =
3
 π +
3
 π =
3
 π

Volume of water left in the cylinder = Volume of cylinder - Volume of solid figure immersed

= π -
π
3
=
π - π
3
 =
π
3
=
3
 ×
22
7
 =
21
 = cm3