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Question

A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of water overflows?


A

0.375

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B

0.625

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C

0.125

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D

0.875

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Solution

The correct option is A

0.375


Radius of the conical vessel, R=AC=6cm

Height of the conical vessel, h=OC=8cm

Radius of the sphere, PD=PC=r

PC=PD=rAC=AD=6 cm

[Since, lengths of two tangents from an external point to a circle are equal]

OCA & OPD are right triangle.
[ Tangent and radius are perpendicular to each other]

OA=OC2+AC2=82+62 =100=10 cmOP2=OD2+PD2

OD=OAAD=106=4 cmOP=OCPC=8r (8r)2=42+r26416r+r2=16+r216r=48r=3 cm.

Volume of water overflown = Volume
of sphere
=43πr3=43π×(3)3=36π cm3

Original volume of water = volume of
cone
=13πr2h=13π×62×8=96π cm3


Fraction of water overflown =Volume of water overflownOriginal volume of water=36π96π=38=0.375

Fraction of water overflown is 0.375


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