A sign somewhere in the United States says that London, England, is 4386 miles away and that Tokyo, Japan, is 6182 miles away.

Where in the U.S. is the sign?

(Consider the earth to be a perfect sphere of radius 3959 miles, and that London is at 51°30’ North, 0°10’ West and Tokyo is at 35°40’ North, 139°45’ East.)

The sign is in Topeka, Kansas.

After a whole day with no comments on this puzzle, I decided to solve it. First I Googled "spherical trigonometry" to learn the first law of cosines for a spherical triangle:
(1) cos a = cos b cos c + sin b sin c cos A
where a, b, and c are the legs of a spherical triangle and A is the angle between b and c. Then, considering the geometry of this problem, I labeled the points of interest as T (Tokyo), L (London), N (north pole), and U (the mystery spot in the US).

For sperical triangle TNU, let a=side TU, b=side TN, and c=side UN. In this case, a=the arc corresponding to an arc length of 6182 miles, or 1.5615054 radians, and b=90 minus the north latitude of T, or 0.9482956 radians. Now we plug these values into (1), where c and A are the values to be found. Repeating this process for spherical triangle ULN, a=arc length for 4386 miles, or 1.1078555 radians and b=90-north lat of L, or 0.6719518 radians. Leg c for this triangle is the same as it was for the other one, and angle A for the two triangles adds up to the difference in longitude between L and T (on the far side of the earth; the US is obviously not between London and Tokyo the short way) which is 220.083333 degrees or 3.8411785 radians.

We now have two equations and two unknowns, but the sines and cosines made it too messy for me to solve algebraically, so I wrote a .vi in LabVIEW (it's all I have, so I used it) to solve numerically.

It turns out c=0.8892906, or 50 degrees 37.2 minutes from the north pole, i.e. 39 degrees 2.8 minutes north latitude. A=2.1741817, or 124 degrees 34.3 minutes east of Tokyo, which is to say 95 degrees 40.7 minutes west longitude. This puts us right in Topeka.

Now that I know spherical trig, I plan never to use it again! lol

Posted by Bryan on 2003-05-06 18:17:26