  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 right−angledΔBAC,We haveBC2=AC2+AB2[∵ Using Pythagoras Theorem]⇒BC2=(8)2+(6)2⇒BC2=64+36=100⇒BC=10 cm Let radius of the sphere be r cm.  Then OA = r cm ⇒OC=AC−OA=(8−r)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 right−angled ΔOCD,OC2=OD2+CD2[∵ Using Pythagoras Theorem]⇒(8−r)2=r2+42[∵ Using Pythagoras Theorem]⇒64+r2−16r=r2+16[∵CD=BC−BD=10−6=4 cm]⇒64+r2−16r=r2+16⇒48=16r⇒r=3 cm ∴ Fractionof water which overflows=Volume of sphereTotal volume of water in cone before immersion=43πr313πR2H=4×(3)36×6×8=38Mathematics

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