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Question
The radii of the bases of two solid right circular cones of same height are r1 and r2 respectively. The cones are melted and recast into a solid sphere of radius R. Find the height of each cone in terms r1, r2 and R.
Solution
Let the height of the solid cones be 'h'
Volume of solid circular cones
`V_1 = 1/3pir_1^2h`
`V_2 = 1/3pir_2^2h`
Volume of sphere
= `4/3piR^3`
Volume of sphere = Volume of cone 1 + volume of cone 2
`4/3piR^3 = 1/3pir_1^2h + 1/3pir_2^2h`
`=> 4R^3 = r_1^2h + r_2^2h`
`=> h(r_1^2 + r_2^2) = (4R)^3`
`=> h = (4R^3)/((r_1^2 + r_2^2)`
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