## Outline Mathematics

Geometry

# Longitude, Latitude and Distance to the Equator

Here's a problem for the sixth grade from the excellent and highly recommended collection Invitation to a Mathematical Festival. The solution to the problem depends implicitly on specific knowledge that students to whom the problem has been offered might or might not possess. What is it?

A helicopter left Moscow and first flew 300 km to the south, then 300 km to the west, 300 km to the north and 300 km to the east, before it finally landed. Did it end up to the south of Moscow, to the north of it or on the same latitude? Did it end up to the east of Moscow, to the west of it, or on the same longitude? Assume the Earth is a perfect sphere for the problem.

### References

- I. Yashchenko, Invitation to a Mathematical Festival, MSRI/AMS, 2013, pp 31, 136

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Copyright © 1996-2018 Alexander Bogomolny

A helicopter left Moscow and first flew 300 km to the south, then 300 km to the west, 300 km to the north and 300 km to the east, before it finally landed. Did it end up to the south of Moscow, to the north of it or on the same latitude? Did it end up to the east of Moscow, to the west of it, or on the same longitude? Assume the Earth is a perfect sphere for the problem.

### Solution

The helicopter flew south along the Moscow meridian,meridian,parallel, then along a parallel,meridian,parallel, then along a meridian,meridian,parallel in the northern,eastern,northern,western,southern direction, and then along a more northern parallel. Since all meridians,meridians,parallels are the same in length, its latitude,longitude,latitude will remain unchanged - it will fly the same number of degrees in the southern direction as in the northern, since equal distances correspond to an equal number of degrees along the circles of the same radius. Yet the parallels are different in length,length,width,height: the further north it is, the shorter it is. Thus on the more northern parallel the same 300 km will correspond to more,fewer,more degrees.

Thus, the helicopter will end up to the east,east,west of Moscow on the same latitude.

### Remark

The problem has been offered at a Moscow olympiad to the sixth grade students who most likely have been aware of Moscow's location in the Northern hemisphere far away from the Equator, although I doubt anybody their age knew the exact distance - 6206 km, which is by far longer than 300 km specified in the problem. However, such a piece of information would not be amiss as without knowing the exact position of Moscow on the globe, the solution, if taken literally, is not correct. For example, had the starting point been Macapá, Brazil (4 km north of the Equator) or Nairobi, Kenya (140 km north of the equator), the answer would have been different: the helicopter would have landed west of where it started. Needless to say that if Moscow was located in the Southern hemisphere the solution, again, would require certain modification.

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Copyright © 1996-2018 Alexander Bogomolny