The volumes of two similar cylinders are in ratio of 8:27. How can I find the ratio of the base radii of the cylinders?

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This is an example of the volume of similar solids. Two solids are regarded similar if and only if they are the same solid and their corresponding linear measurements, namely radii, heights, base lengths are proportional. 

Since we are dealing with the volume of two similar cylinders, we need to know what the scale factor is or the volume ratio between the two similar cylinders. If two similar solids have a scale factor `a/b`  , then the volumes of the similar cylinders has a volume ratio: `(a/b)^3`   .

In the above example we are given the volume ratio, and we need to find the base radii ratio of the cylinders. 

The base radii is a linear measurement, where as volume is a cubic function. In order to find the base radii from the volume ratio, we need to find to cube root the base radii. This is done as follows: 

`(a/b)^3 = (8/27)`

`(a/b) =root(3)(8/27)`

`(a/b) = 2/3`

Therefore the ratio of the base radii is 2/3. can be written as 2:3.

 

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The complicated way is to write out the ratio of the volumes, set equal to 8/27 and solve for the ratio of the radii.

The easy way is to note that the figures are similar. For similar figures all corresponding lengths are in the same ratio called the scale factor. (All corresponding lengths -- radii, circumference, diameter, altitudes, etc...) 

Let the scale factor be a:b.

Then all corresponding areas are in the ratio a^2:b^2.

All corresponding volumes are in the ratio a^3:b^3.

Here the volumes are in the ratio 8:27, so the scale factor is 2:3. (The cube root of 8 to the cube root of 27.)

The ratio of the radii is 2:3.

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