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Question

A solid consisting of a right 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, if the radius of the cylinder is 60cm and its height is 180cm, the radius of the hemisphere is 60cm and height of the cone is 120cm, assuming that the hemisphere and the cone have common base.

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Solution

For the cylinder we have
Radius of the base=r=60cm
Height =h=180cm

Volume of water that the cylinder contains =V=πr2h={π×602×180}cm3

For conical part, we have
Radius of the base =r=60cm
Height =h1=120cm

Volume of conical part V1=13πr2h1=13π×602×120cm3={π×602×40}cm3
For hemispherical part, we have
Radius =r=60cm

Volume of the hemisphere V2={23π×603}cm3

V2={2π×602×20}cm3=(40π×602)cm3

Hence volume of the water left out in the cylinder

V=V1V2={π×602×180π×602×4040π×602}cm3=π×602×{1804040}cm3

=227×3600×100cm3

=22×36700m3

=1.1314m3

1035678_1010814_ans_fb121d8ccd464e198db8b6bc8bc41337.png

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