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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 in the water and its size is such that when it touches the sides, it is just immersed.  What fraction of the water overflows? [4 MARKS]


Solution



Radius of conical vessel, R = 6 cm and height of conical vessel, H = 8 cm
In rightangledΔBAC,We haveBC2=AC2+AB2[ Using Pythagoras Theorem]BC2=(8)2+(6)2BC2=64+36=100BC=10 cm
Let radius of the sphere be r cm.  Then OA = r cm

OC=ACOA=(8r)cmAlso,Angle 1=90[Radius is perpendicular to the tangent]andAB=BD=6 cm[Tangents drawn from external]  [point to a circle are equal]In rightangled ΔOCD,OC2=OD2+CD2[ Using Pythagoras Theorem](8r)2=r2+42[ Using Pythagoras Theorem]64+r216r=r2+16[CD=BCBD=106=4 cm]64+r216r=r2+1648=16rr=3 cm
Fractionof water which overflows=Volume of sphereTotal volume of water in cone before immersion=43πr313πR2H=4×(3)36×6×8=38

Mathematics

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