A man starts a journey by driving directly South for 30 km, then directly East for 40 km, and finally directly North for 60 km. What is the shortest distance between the points where he stopped and where he started?
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use the pythagorean theorem – Gregory Grant Aug 01 '16 at 20:28
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1Is it on the surface of the Earth? And does the final shortest distance has the same meaning as the "drive" distance? – z100 Aug 01 '16 at 20:34
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1@z100 Lets discuss here instead of in the other answer. I'm also thinking about this. If your sphere has the "right" size it is easy. Take a sphere s.t. the latitude 45 km south of the north pole has circumference 80 km. Start 15 km south of the north pole. Then your path takes you back to where you started. – Maik Pickl Aug 01 '16 at 20:46
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@Maik Pickl Ok, it is just a variant of a well known "polar bear problem". – z100 Aug 01 '16 at 20:49
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@z100 I just realized that strictly speaking the last kilometers on this path will not be "directly north". – Maik Pickl Aug 01 '16 at 20:55
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Yes, you are right @z100 here is a reference among others to this classical problem. Nevertheless, knowing that things should happen around the North Pole is not all... – Jean Marie Aug 01 '16 at 21:12
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Hint:
Supposing that the journey is in a plane (that is a good approximation of the Earth surface for such distances), starting from the point $(0,0)$ the journey can be represented as: $$ (0,-30)+(40,0)+(0,60)=(40,30) $$ Can you see why? And can you find the distance of the point $(40,30)$ from $(0,0)$?
Emilio Novati
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1Thats assuming the journey takes place on a plane. If you are on a sphere that is big enough you could start 30 km north from a point where the latitude is such that one time going around is 40 km (I hope you know what I mean) and then going north again takes you 30 km away from your starting point. – Maik Pickl Aug 01 '16 at 20:36
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Yes. I suppose that the journey is in a plane. I add to the answer. – Emilio Novati Aug 01 '16 at 20:41
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@Maik Pickl Possibly one can return to the same point if when the direction is chosen it holds till the chosen distance is reached. So somwhere near the north pole go 30 km south, then 40 km exactly to the oposite meridian point and then turn to the north pole- well last 30 km will be to the south but without diretion change. – z100 Aug 01 '16 at 20:42
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I am sorry, @Emilio Novati, I disagree : assuming that the problem can be transformed into a plane problem is not adequate if the deformation is too big in terms of longitudes ; said otherwise if the solution takes place around the singularity constituted by the north pole (there is a discontinuity there for geographical coordinates ; in terms of differential geometry, a sphere cannot be represented as a variety using an atlas with a single map). And, most probably it is around North Pole that the minimum is reached (see the "polar bear problem" with the reference I have given) – Jean Marie Aug 01 '16 at 21:33