12th Putnam 1952

Problem A4

A map represents the polar cap from latitudes -45o to 90o. The pole (latitude 90o) is at the center of the map and lines of latitude on the globe are represented as concentric circles with radii proportional to (90o - latitude). How are east-west distances exaggerated compared to north-south distances on the map at a latitude of -30o?



Answer: too high by a factor 4π/√27 = 2.42.

Let the globe have radius R. Then the distance from the pole to the circle of latitude at -30 is 2πR/3. The circumference of the circle is 2πR (√3)/2. The ratio of circumference to distance is (√27)/2. On the map the corresponding ratio is 2π. Thus the map overstates by a factor 4π/√27.



12th Putnam 1952

© John Scholes
5 Mar 2002