Question

A solid, consisting of a right circular cone standing on a hemisphere, is placed upright, in a right circular cylinder, full of water and touches the bottom. Find the volume of water left in the cylinder, having 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. Give your answer to the nearest cubic centimetre.

Answer 1

Radius of cylinder = 3 cm

Height of cylinder = 6 cm

Radius of hemisphere = 2 cm

Height of cone = 4 cm

Volume of water in the cylinder when it is full

= πr²h

= π × 3 × 3 × 6

= 54π cm³

Volume of water displaced = Volume of cone + Volume of hemisphere

= `1/3 pir^2h + 2/3 pir^3`

= `1/3 pir^2 (h + 2r)`

= `1/3 pi xx 2 xx 2(4 + 2 xx 2)`

= `1/3 pi xx 4 xx 8`

= `32/3 pi cm^3`

Therefore, volume of water which is left

= `54 pi - 32/3 pi`

= `130/3 pi cm^3`

= `130/3 xx 22/7 cm^3`

= `2860/21 cm^3`

= 136.19 cm³ 

= 136 cm³