- Paper bag problem
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 acushion orpillow , made out of two pieces of material which can bend but not stretch.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::$V=w^3\; left\; (h/\; left\; (pi\; w\; ight\; )\; -0.142\; left\; (1-10^\; left\; (-h/w\; ight\; )\; ight\; )\; ight\; ),$

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

(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).

**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:

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

or roughly 0.19. According to

Andrew Kepert at theUniversity 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.

**References***

* cite journal

author=Baginski, F.; Chen, Q.; and Waldman, I.

year=2001

title=Modeling the Design Shape of a Large Scientific Balloon

journal=Applied Mathematical Modelling

volume=25

pages=953–956

doi=10.1016/S0307-904X(01)00024-5

* cite journal

author=Mladenov, I. M.

year=2001

title=On the Geometry of the Mylar Balloon

journal=C. R. Acad. Bulg. Sci.

volume=54

pages=39–44

* cite journal

author=Paulsen, W. H.

year=1994

title=What Is the Shape of a Mylar Balloon?

journal=American Mathematical Monthly

volume=101

pages=953–958

doi=10.2307/2975161

* cite journal

author=Anthony C Robin

year=2004

title=Paper Bag Problem

journal=Mathematics today ,Institute of Mathematics and its Applications

id=ISSN|1361-2042

volume=June

pages=104–107**External links*** [

*http://www.ics.uci.edu/~eppstein/junkyard/teabag.html The original statement of the teabag problem*]

* [*http://maths.newcastle.edu.au/~andrew/teabag/ Andrew Kepert's work on the teabag problem*]

* [*http://frey.newcastle.edu.au/~andrew/teabag/folding/curvedFold.html Curved folds for the teabag problem*]

* [*http://www.dse.nl/~andreas/teabag.html A numerical approach to the teabag problem by Andreas Gammel*]

* [*http://mathworld.wolfram.com/PaperBag.html MathWorld article*]

*Wikimedia Foundation.
2010.*

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