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Math: How to find the possible values of r in a torus, using the volume equation?
I have this Math problem in which requires me to find the possible values for r:
The volume of a torus is give by the formula V=2 * π^2 * r^2 * R where r and R are the radii shown and r is equal or less than R.
A metal ring in the shape of a torus has a volume of 100 cubic centimeters. Choose three possible values of r, and find the corresponding values of R.
I don't know how to find the possible values of r. Many of the numbers I have randomly tried give the result that r is greater than R, which is incorrect. Could you please make it clear for me in this problem?
Thank you very much!
1 Answer
- Scarlet ManukaLv 78 years agoFavourite answer
Like most problems, it helps if you think about things a little bit before rushing in.
π² is roughly 10, so if we want V = 100 = 2π² r² R, we're going to need r² R to be around 5 or so. So you can't choose e.g. r = 2 cm, since then you'd be looking at R around 1.25; you need smaller values of r.
Now, to actually finding R: we already said 100 = 2π² r² R, and it's pretty easy see that we can rearrange that to give R = 100 / (2π² r²) = 50 / (π² r²).
So choose some values of r:
r = 1 cm => R = 50 / π² = 5.07 cm (3sf)
r = 0.5 cm => R = 50 / (π² (0.5)²) = 20.3 cm (3sf)
r = 1.5 cm => R = 50 / (π² (1.5)²) = 2.25 cm (3sf)
With a little thought you should be able to see that the value you choose for r must be less than the cube root of 50 / π², i.e. 1.717 cm (4sf), in order to have R > r. Any positive number less than this will be fine.
