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I think you want to rephrase it as, "What geometric surface encloses the maximum volume with a given surface area". In other words, you have 1 unit-squared of material and you want to enclose a surface to yields maximum volume.How do I set up the following problem?
What geometric surface encloses the maximum volume with the minimum surface area?
I don't think I want to rephrase it. The question is number 13 hereI think you want to rephrase it as, "What geometric surface encloses the maximum volume with a given surface area". In other words, you have 1 unit-squared of material and you want to enclose a surface to yields maximum volume.
Let $\Omega$ be bounded region in $\mathbb{R}^3$. You want to maximize,
$$ \iiint_\Omega 1 $$
Given the condition that,
$$ \iint_{\partial \Omega} 1 ~ ds = 1 $$
If we care about small details this question is a lot more complicated to phrase. Then we need to restrict ourselves to measurable sets such that .. blah blah blah.
Then how would I show a sphere is the smallest surface containing a given volume?That's still a very badly phrased question. I think they expect you to combine the facts that the smallest surface containing a given volume is a sphere and that the largest area for a given size surface is a sphere.