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Simon from Elizabeth College, Guernsey and Andrei from Tudor Vianu National College, Romania have both solved this problem and both solutions are used below.
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To solve this problem we shall consider that the point is
situated at latitude \alpha. To travel around the line of
latitude, the distance from P to Q would be half the circumference
of the circle at latitude \alpha. This circle has a radius R
\cos \alpha, where R is the radius of Earth.
So, the distance traveled from P to Q on the line of latitude
is d_{lat} = \pi R \cos \alpha.
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Traveling over the line of longitude, the circle on which we
have to calculate the distance is a great circle of the sphere, and
the angle of displacement is 2(\pi /2 - \alpha) radians. The
distance is therefore 4\pi R(90-L)/360 where L is the angle of
latitude in degrees or equivalently d_{long} = 2R (\pi/2 -
\alpha).
It is clear that the path on a great circle is always
shorter.
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A graph of the ratio of these distances shows that this ratio
seems to tend to \pi /2 as the line of latitude approaches the
pole, that is the ratio {d_{lat}\over d_{long}}= {\pi R \cos
\alpha \over 2R (\pi/2 - \alpha)} tends to a limit as \alpha \to
\pi/2.
This ratio can also be written as {d_{lat}\over d_{long}}=
{\pi R \sin (\pi/2- \alpha) \over 2R (\pi/2 - \alpha)}. As we
know {\theta\over \sin \theta} \to 1 as \theta \to 0 we can
take \theta = \pi/2 - \alpha and we see that this limit is \pi
/2.
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