Folding a Cup

Yesterday, there was a TV quiz in Germany „How clever are the Germans?“. I regret to have missed it. However, you can currently see the complete show on-line (zdf.de). Note that due to German TV regulations, it must go offline soon. I was very disappointed with the knowledge level, which manages to be below the guessing level at some questions. But see yourself! This is an international blog, so be warned that the language is German.

However, there are some other questions on-line (zdf.de), and of course there are mathematical questions among them. Try yourself, and see how you do.

The question, which month is the longest in Germany is clever, since most people (including myself) might miss the switch from summer time to GMT taking place in October, which adds an extra hour. Others questions were silly, like the question how high a man could jump „with the force of a flea“. The correct answer seems to me that he cannot jump at all. What they mean is: If you magnify a flea to the size of a man, how high would the jump magnify (unchecked answer is about 150m). By the way, a surprising observation with a mathematical background is that all animals that jump can jump almost the same height of about half a meter.

However, there was one question, which attracted my interest. If you make two cups from a paper, one with a square base and one circular base, which one is larger? The exact phrase was „deform“ the area of a paper. If you put it like this, the problem sounds equivalent to the exercise to maximize the volume with given surface. Doing this for for the cuboid formed cups and for cylinder cups, we get that the cylinder cup is a bit larger. It has to be twice as wide than high, which is not really practical, by the way.

First problem: Verify this!

However, if we actually want to make a cup, things are different. We probably do not want to glue snippets of paper together to a circular form. So let us stick with very simple things, like one square cut out from a sheet of paper, and we cut out our circle from that square, plus some stripes of paper forming the upward walls of the cup. I did the computations for a DIN paper with edges 1 and square root of 2, and it seems that the cup with the square base can be made a bit larger in this case.

Second problem: Verify that!

It seems, things in real life math are never as easy as our school exercises.

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