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Topic: Cooking. Fast!
Replies: 0

 Rouben Rostamian Posts: 16 Registered: 12/6/04
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 off-center 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 non-heated dead area in its center.) Clearly, you won't
place the tiny pot in the dead core area; an off-center 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