Proportions of Exact EnlargementsDate: 03/18/98 at 03:57:18 From: Emma J. Subject: Ratio of 2 cylinders A two-liter can is an exact enlargement of a one-liter can. Explain why their base radii are in the ratio 1:(the cube root of 2). I have no idea how to explain this conclusion and would be grateful for some help. Thanks. Date: 03/18/98 at 09:56:28 From: Doctor Sam Subject: Re: Ratio of 2 cylinders Emma, "Exact enlargement" is an informal way of saying that the two cans are similar. Similar figures, whether they are 2D-plane figures or 3D- shapes, are proportional. A general principle that can be proved with a little geometry relates the volumes, areas, and corresponding lengths of similar figures. Here is the principle. I'll follow it up with a few examples to try to make it meaningful. In similar figures, corresponding lengths are proportional (that is, have the same ratio). In similar figures, corresponding areas are proportional to the squares of corresponding lengths. In similar figures, corresponding volumes are proportional to the cubes of corresponding lengths. For example, take a 3 x 5 rectangle. If you enlarge it with a photocopier, you might get a rectangle that is 6 x 10 or 9 x 15. Corresponding sides have the same ratio: 6/3 = 10/5; 9/3 = 15/5. The areas of these rectangles are 15, 60, and 135 respectively. The ratio of areas: 15/60 = (3/6)^2. Note that 15/60 = 1/4 and 3/6 = 1/2 and (1/ 2)^2 = 1/4. Another example: take a 2 x 2 x 2 cube. It is similar to a 5 x 5 x 5 cube. The ratio of corresponding sides is 2/5. The ratio of areas of corresponding faces is 4/25 = (2/5)^2. The ratio of their volumes is 8/125 = (2/5)^3. In your example, your cylinders are similar. The ratio of their volumes is 2/1. Corresponding lengths (the radii of their bases or their heights) will have ratio (a/b) and (a/b)^3 = 2/1. That should be enough background to show the relationship that you want. -Doctor Sam, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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