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Question 1139254: The radii of two spheres are in a ratio of 1:8. What is the ratio of their volumes?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Answer is 1:512
What does this mean exactly? It means that the first sphere might have a volume of 1 cubic inch, so the second sphere would have a volume of 512 cubic inches (which is 512 times that of the first volume). Or the first sphere's volume is 2 cubic inches while the second sphere has a volume of 2*512 = 1024 cubic inches. The individual volumes don't matter as long they fit the proper ratio.
To get this answer, you simply cube each part
1^3 = 1*1*1 = 1
8^3 = 8*8*8 = 512
That's how I went from 1:8 to 1:512. Or you can think of cubing the ratio 1:8 to have it turn into (1:8)^3 = 1:512, though this notation might be confusing to deal with.
Here's one way to think about why this works: Consider that you have a wooden block that is 1 inch by 1 inch by 1 inch. The volume is therefore 1*1*1 = 1 cubic inch. Now let's add in another wooden block that is 8 by 8 by 8, so its volume is 8*8*8 = 512. We can see that the side lengths of the cubes form the ratio 1:8, and in addition, the volumes form the ratio 1:512. This idea for cubes can be extended to any 3D shape because we can sub-divide the 3D object into tiny cubes (of course this will most likely be an approximation).
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