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Question
If a sphere of radius 2r has the same volume as that of a cone with circular base of radius r, then find the height of the cone.
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Solution
In the given problem, we are given a cone and a sphere which have equal volumes. The dimensions of the two are;
Radius of the cone (rc) = r
Radius of the sphere (rs) = 2r
Now, let the height of the cone = h
Here, Volume of the sphere = volume of the cone
`(4/3)pi r_s^3 = (1/3)pi r_c^2 h`
`(4/3)pi (2r)^3 = (1/3) pi (r)^2 h`
`(4/3) (8r^3)=(1/3)r^2 h`
Further, solving for h
`h = ((4)(8r^3))/((r^2))`
Therefore, the height of the cone is 32r.
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