Assume earth to be a perfect sphere. Imagine a point P such that, a person starting from P, travelling 1km South, 1km West and 1km North in this order, ends up at the same point P. How many such points P exist on earth? (Exclude the poles) Draw an image to describe your answer.
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Answer:
NOTE: These calculations are based on the assumption that the Earth is a perfect sphere with a circumference of exactly 40,000 km, and therefore a radius of exactly (20000/π) km, which is approximately equal to 6336.1977237 km.
One point P is the North Pole (latitude 90°N). If the person, whom I'll call J, starts at the North Pole and walks southward to a point that is 1 km south of the North Pole, then (s)he will arrive at a latitude which is 1 km south of the North Pole (approximately 89.991°N). S(he) then walks westward for 1 km, which means that (s)he) walks along the circle of latitude which is 1 km south of the North Pole. Therefore (s)he is still 1 km south of the North Pole after (s)he has walked 1 km westward. So after (s)he then walks 1 km northward, (s)he will be back at his starting point at the North Pole. So the North Pole is one point P. The other (infinite number¹ of) points are all located close to the South Pole.