Tokyo, Japan and Tehran, Iran, lie on a latitude of 38 and 47/60 degrees and longitu...

Question

Tokyo, Japan and Tehran, Iran, lie on a latitude of 38 47' 00" and longitude of 141 2' 00" East and 52 5' 00" East, respectively. What is the length of the line of constant latitude between these two cities? Answer in miles.

 

I have tried nearly everything to answer this problem. I have used all the Great Circle Equations, the Haversine Function, etc..., and always seem to be about 30 miles short of the answer, which is supposed to be 4.79*10^3. For the problem set I am doing, that is an unacceptable error. Please tell me what I am doing wrong. Thank you!

Answer 1

If you model the Earth as a sphere with radius R then the circumference of the circle at the equator is 2*pi*R.  A "line of constant latitude" will form a circle with a smaller radius - think of slicing the sphere across at the given latitude.  If you draw a picture you will see that the radius of the circle along a line of constant latitude is R*cos(LAT).  So the circumference of the circle along a line of constant latitude is 2*pi*R*cos(LAT).  The two cities are separated by

(141 degrees 2') - (52 degrees 5') = (88 degrees 57') = 88.95 degrees  so the length of the arc between them is 

(88.95 / 360) of the full circle.  If you use R = 3959 miles you will get  

 

D = 2*pi*3959*cos(38.783)*(88.95/360) = 4791 = 4.79*10^3 miles