Proportions of Exact Enlargements

Date: 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
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