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Question

The radius of a solid metallic sphere is r. A solid metallic cone of height h has base radius r. The two are melted together and recast into a solid right circular cone with base radius r. Prove that the height of the resulting cone is 4r+h.

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Solution

Volume of resulting cone = Volume of sphere + Volume of cone which are melted.
Volume of a sphere of radius r=43πr3
Volume of a cone =13πr2h
where, r is the radius of the base of the cone and h is the height.
Let the height of the resulting cone be H.
Hence, 13πr2H=43πr3+13πr2h
=>H=4r+h
Hence, height of the resulting cone is 4r+h

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