I'd like a formula for a bag made of two flat equal sized rectangles (e.g. a freezer bag). Assume no stretching, and perfect flexibility. Volume in terms of a and b, the dimensions of the bag when flat.
Thanks
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$\begingroup$ It's not homework. I haven't tried anything, it's beyond me but I'm guessing could be done using some combination of spheres and cones. A good approximation would be fine, say +/- 5% $\endgroup$– JodesJul 5, 2012 at 12:19
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1$\begingroup$ As a starting point, a sphere maximises volume for a given surface area, so if we define $r$ to be such that $4\pi r^2 = 2ab$, we can't possibly do better than $\frac{4}{3}\pi r^3$. I suspect that in general this is a very loose bound. $\endgroup$– Ben MillwoodJul 5, 2012 at 12:29
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1$\begingroup$ $\sqrt\frac{ab}{2\pi} = r$, so an upper bound is $\frac{4}{3}\pi(\frac{ab}{2\pi})^\frac{3}{2}$ which simplifies to $\frac{1}{3}\sqrt{\frac{2}{\pi}a^3b^3}$. I suspect this problem of being hard. $\endgroup$– Ben MillwoodJul 5, 2012 at 12:35
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1 Answer
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I think this problem is difficult. It certainly is difficult if you started with two congruent disks rather than two congruent rectangles. The disks problem is sometimes known as the Mylar Balloon problem. See, e.g., "The Mylar Balloon Revisited," 2003, Amer. Math. Monthly (JSTOR link).
See also Igor Pak's 2006 paper, "Inflating polyhedral surfaces" (CiteSeer link).
answered Jul 5, 2012 at 14:49