Shyam travels 7 km toward North, then he turns to his right and walks 3 km. He again turns to his right and moves 7 km forward. Now in which direction is he from his starting point?
1 Answer
Simple answer: East of his starting position.
Actual answer: Depends on how far from the north pole he starts from.
Explanation:
There are a few possible answers here. I'm assuming that both turns are 90 degree turns, the circumference of the Earth at the equator is 40075 km and that the Earth is a perfect sphere (it is not, but calculating on an oblong spheroid would be waaaay too hard).
Starting from most places on Earth the answer is pretty simple. Shyam will end up some distance to the east of his starting position.
As the starting point gets closer to the north pole weird things start to happen.
At exactly 7.951 km south of the north pole, he will end up neither east nor west of his starting position as the circumference 7 km north of that point is exactly 6 km, so he will end up on the "opposite" side of the Earth.
Starting a bit further north, he will end up to the west of his starting position, up to the point where the circumference is 3 km, exactly 7 km north of 7.476 km from the pole. From that distance his endpoint will be his starting point.
This pattern of ending east until he is exactly oposite, then ending up west until he is exactly at the start again will repeat until he starts at exactly 7 km from the north pole. (ending opposite when the circumference is 2 km, at the start when it is 1.5 km, opposite when it is 1.2 km, at the start when it is 1 km, so on and so forth)
From exactly 7 km from the north pole, his ending position will be somewhat northwest from his starting position, since when he reaches the north pole and he turns right he starts walking south, then the second turn makes him start to walk west. At that point, 3 km from the pole, the circumference is only 18.9 km, so his 7 km walk to the west will take him almost to the opposite side of that circle, but still, he will cross the original line and go west of it.
North of that he will cross the north pole before finishing his first 7 km walk, so his turning right will lead him to walk west, and his second turn right will lead him north again. This is pretty much the same case as the one starting between 7.951 km and 7 km from the north pole.
The last "weird" position Shyam could start from is the exact south pole. From there his ending position will be exactly where he started.
In case you want to play with these numbers and verify them, this is how to do it:
To calculate the circumference at any latitude (assuming a sphere) you multiply the circumference at the equator by the cosine of the latitude:
To calculate the distance from the north pole where a certain circumference happens you invert that formula. First you calculate the latitude by using the arc cosine (
Once you get the latitude, plug it into this site, Distance between latitude/longitude calculator and input 90 degrees latitude in the "from" field and the latitude you got from the calculation above in the "to" field.
