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Over the Pole

Age 16 to 18
Challenge Level

Simon from Elizabeth College, Guernsey and Andrei from Tudor Vianu National College, Romania have both solved this problem and both solutions are used below.

line of latitude
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 $d_{lat} = \pi R \cos \alpha$.

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.
table of ratios

graph of ratios
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 $\pi /2$.

Andrei calculated this limit using L'Hopital's Rule (case $0/0$): $$\lim_{\alpha\to \pi/2}{\pi R\cos \alpha \over R(\pi -2\alpha)} = \lim_{\alpha\to \pi/2}{\pi \cos \alpha \over (\pi -2\alpha)} = \lim_{\alpha\to \pi/2}{-\pi \sin \alpha \over -2} = {\pi \over 2} \approx 1.57$$