If the radius of the base of a cone is doubled and the volume of the new cone is three times the volume of the original cone, then what will be the ratio of the height of the original cone to that of the new cone?

The volume of a cone equals one-third times pi times the square of the radius times the height. Let the initial radius be r and height be h, so the original volume is one-third pi r squared h; the new volume with doubled radius is three times one-third pi r squared h, which equals pi r squared h. Equating this to one-third times pi times the square of two r times the new height H gives pi r squared h equals four-thirds pi r squared H. Solving the ratio h to H yields 4 to 3.

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