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Cooking. Fast!
Posted:
Feb 18, 1993 9:29 PM


Here is geometry problem motivated by a very basic urge  Hunger!
I have an electric stove in my kitchen. You know, the kind with flat spiraling coils that get hot when you turn them on, and you place a pot or frying pan on them for cooking.
Now I have to admit that I am a rather impatient person, and when I want to heat something on the stove, I want it done FAST! Here comes my dilemma: How does one position a pot on the stove for fastest results?
If the pot is large enough, so that its bottom can completely cover the heating coil, the choice is clear: place the pot centered with the coil and hold your breath!
The problem arises when the pot is small, so that it will not cover the coil. Placing it centered with the coil leaves the coil exposed around its edges, where precious heat is lost to the surrounding air instead of heating the pot. The coil is a lot more effective in heating the pot when it is in direct contact with the bottom of the pot! Perhaps placing the pot offcenter provides a larger contact area? You can even imagine an extreme case, when you use a pot so small that it fits right in dead area in the center of the coil. (I forgot to say at the beginning that the heated area of a cooking coil is more or less in the shape of an annular region, with a significant nonheated dead area in its center.) Clearly, you won't place the tiny pot in the dead core area; an offcenter position will provide a better contact with the heated coils and will give faster results.
I hope that the above is a good motivation for the following geometry problem:
Problem: Consider an annular region bounded by concentric circles of radii a and b. Consider a disk of radius R that can slide in the plane with respect to the annulus. Determine the position of the disk with respect to the annulus such that the overlap area of the disk and annulus is as large as possible.
Note 1: The answer is quite simple and useful. In fact, I always position pots on my stove according to the solution to this problem.
Note 2: I posted this puzzle on usenet's sci.math newsgroup several years ago as a challenge. A few people actually went as far as solving and posting their solutions.
Note 3: I may even post my solution to this newsgroup at some later time, but I would rather see other people's attempts and approaches to the problem.
 Rouben Rostamian



