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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.
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Solution
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
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