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Question

The height of a right circular cone is 20 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 18 of the volume of the given cone, then at what height above the base is the section made? [HOTS] [CBSE 2014]

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Solution



We have,Height of the given cone, H=20 cmLet the radius of the given cone be R,the height of the smaller cone be h andthe radius of the smaller cone be r.Now, in AQD and APC,QAD=PAC Common angleAQD=APC=90°So, by AA criteriaAQD~APCAQAP=QDPChH=rR .....iVolume of smaller cone=18×Volume of the given coneVolume of smaller coneVolume of the given cone=1813πr2h13πR2H=18rR2×hH=18hH2×hH=18 Using ihH3=18hH=183h20=12h=202h=10 cm PQ=H-h=20-10=10 cm

So, the section is made at the height of 10 cm above the base.

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