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In geometry, the paper bag problem or teabag problem involves calculating the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, made out of two pieces of material which can bend but not stretch. ## The square teabag

In the special case where the bag is sealed on all edges and is square with unit sides, h = w = 1, and so the first formula estimates a volume for this of roughly: ## References

The problem is made even more difficult by assuming that the bag is made out of a material like paper or PET film which can neither stretch nor shear.

According to Anthony C. Robin, an approximate formula for the capacity of a sealed expanded bag is:

- $V=w^3\; left\; (h/\; left\; (pi\; w\; right\; )\; -0.142\; left\; (1-10^\; left\; (-h/w\; right\; )\; right\; )\; right\; ),$

where w is the width of the bag (the shorter dimension), h is the height (the longer dimension), and V is the maximum volume.

A very rough approximation to the capacity of a bag that is open at one edge is:

- $V=w^3\; left\; (h/\; left\; (pi\; w\; right\; )\; -0.071\; left\; (1-10^\; left\; (-2h/w\; right\; )\; right\; )\; right\; )$

(This latter formula assumes that the corners at the bottom of the bag are linked by a single edge, and that the base of the bag is not a more complex shape such as a lens).

- $V=frac\; 1\; \{pi\}\; -\; 0.142\; cdot\; 0.9$

or roughly 0.19. According to Andrew Kepert at the University of Newcastle, Australia, the upper bound for this version of the teabag problem is 0.217+, and he has made a construction that appears to give a volume of 0.2055+.

In the article referred to above A C Robin also found a more complicated formula for the general paper bag. Whilst this is beyond the scope of a general work, it is of interest to note that for the tea bag case this formula gives 0.2017, unfortunately not within the bounds given by Kepert, but significantly nearer.

- Baginski, F.; Chen, Q.; and Waldman, I. (2001). "Modeling the Design Shape of a Large Scientific Balloon".
*Applied Mathematical Modelling*25 953–956. - Mladenov, I. M. (2001). "On the Geometry of the Mylar Balloon".
*C. R. Acad. Bulg. Sci.*54 39–44. - Paulsen, W. H. (1994). "What Is the Shape of a Mylar Balloon?".
*American Mathematical Monthly*101 953–958. - Anthony C Robin (2004). "Paper Bag Problem".
*Mathematics today, Institute of Mathematics and its Applications*## External links

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Last updated on Thursday September 04, 2008 at 05:42:32 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday September 04, 2008 at 05:42:32 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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