Monday, June 29, 2009
Temperature twins
There should be some mathematics on this blog :-) I am still waiting for a solution to this question I asked in class: Modeling the earth as a sphere, and assuming temperature varies continuously on its surface, show there are two antipodal (i.e. opposite) points on the Earth's surface with the same temperature. (Two students came close but didn't push through to the end.)
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Basically, what you've got here is an application of the Intermediate Value
ReplyDeleteTheorem. It says that if you've got a continuous function f, and two
different input values (call them a & b) for the function that give you two
different output values (call them v & w) for the function, then for every
output value you pick between v & w, there's some input value between
a & b that you can plug into f to get that output value. In other words,
everything between v & w is hit by f.
Take two diametrically opposed,
or "antipodal", points, and define a function f that gives you the difference
between their temperatures. For instance, if we plug in the north pole and
the south pole, we'll get a difference of -30 degrees or something, and if
we plug in the south pole and the north pole (the order is important in
subtraction) we'll get 30 degrees, or whatever the opposite of the first
value was.
For the sake of argument, let's say that the temperatures at
the north pole and the south pole are different. Now pick a continuous
path that goes from the north pole to the south pole. It doesn't matter
which one you pick, because we'll eventually be picking a bunch.
Now travel along that path. When you start out at the north pole, your
value for f is on one side of zero, and when you end up at the south pole,
it's on the other side of zero. So somewhere in the middle there, it had to
be zero. i.e THE INTERMEDIATE VALUE THEOREM!! So
what we've found is a pair of antipodal points with the same temperature,
because the difference between them is zero.
Now pick another path, and do the same thing. You'll get another pair of
antipodal points like that, that have the same temperature. If you keep
doing that, with lots of different paths, you'll eventually get a whole
band's worth of such points. When I say band, I mean that there is a whole
closed curve (a loop on the surface of the Earth) that contains only this
kind of points, points where the temperature on the other side of the earth
is exactly the same. Also,if a point is on this curve,
its antipodal point.
The solution in my above comment is from:
ReplyDeleteAsk Dr. Math
http://mathforum.org/library/drmath/view/54746.html