12th Putnam 1952 Problem A4 A map represents the polar cap from latitudes -45 o to 90 o . The pole (latitude 90 o ) is at the center of the map and lines of latitude on the globe are represented as concentric circles with radii proportional to (90 o - latitude). How are east-west distances exaggerated compared to north-south distances on the map at a latitude of -30 o ? Solution
: 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 jscholes@kalva.demon.co.uk 5 Mar 2002