Relation Between Lengths of Solids With Volumes in a Certain Ratio

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If the volumes of two solid is in a certain ratio, we can often find relationships between some lengths of the solids even if the solids are not similar.

Suppose the volume of the cylinder and sphere below are in the ratio 2:3. The radius of the sphere is twice the radius of the cylinder, or equivalently, the radius of the cylinder is half the radius of the sphere. If the radius of the sphere isfind the height of the cylinder in terms of

The volume of the sphere is

The volume of a cylinder of radiusand heightisso the volume of the cylinder above is

The ratio given in the question means thatso

Then

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