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Math Help - Surface Area to Volume Ratio

  1. #1
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    Surface Area to Volume Ratio

    It's intuitive to me that for a given volume, a sphere will have the smallest surface area to volume ratio. But how do we prove it?
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  2. #2
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    Three Hints:

    1) Since the volume is fixed, there is no need to theorize on the ratio. Examining the Surface Area will be sufficient.

    2) Have you considered the examination of one dimension less? For a given area, a circle presents the least circumference.

    3) You may need to experiment with different shapes.
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  3. #3
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    1) obviously

    2) Yes, that is true and yet I don't have any idea on how to prove it

    3) What do you mean?
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  4. #4
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    Perturb your sphere and prove that anything else makes it bigger.

    Flatten it.
    Make corners on it.
    Dent it in.
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  5. #5
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    Quote Originally Posted by TKHunny View Post
    Perturb your sphere and prove that anything else makes it bigger.

    Flatten it.
    Make corners on it.
    Dent it in.
    I have 2 questions:

    1) Exactly what kind of maths would I need to use to "perturb" the sphere?

    2) Even if I perturb the sphere and reach the conclusion that any small changes lead to increased surface area, how do I know that the spherical shape isn't a "local minimum"?
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  6. #6
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    1 ) "Calculus of variations"
    http://mathworld.wolfram.com/CalculusofVariations.html

    2) ?? You don't know "the spherical shape isn't a "local minimum"; it is!
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  7. #7
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    1) Yep, as I thought, something I don't know, I'll have to study that

    2) I meant - how do I know that it is not only a local minimum but also the global minimum?
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