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Question
A metallic sphere of radius 10.5 cm is melted and thus recast into small cones, each of radius 3.5 cm and height 3 cm. Find how many cones are obtained.
Solution
The radius of sphere r = 10.5 cm
The volume of sphere
`= 4/3 pi x^3`
`= 4/3 pi xx (10. 5)^3`
`= 4/3 pi (21/2)^3`
`= 4/3 pi (441 xx 21)/8`
`=(144 xx 21)/6 pi (cm)^3`
Let n be the number of cones obtained when the sphere is recast in to small cones, each of radius 3.5 and height 3 cm.
Then, volume of sphere = n × volume of cone
`(441 xx 21)/6 pi = n xx 1/3 xx pi xx (3.5)^2 xx 3`
`(441 xx 21)/6 = n xx 1/3 xx 7/2 xx 7/2 xx 3`
`441 xx 21 = (n xx 49 xx 3)/2`
`n = (441 xx 21 xx 2 )/(49 xx 3) = 126`
`n = 126`
Hence, the no. of cones = 126
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