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NCB  Deposit #8
  contributed by

Dr. Frank Wattenberg, United States Military Academy, West Point, NY

  Estimating the Distance between Two Locations
Estimating the Area of Colombia
Using Map Scales and the Pythagorean Theorem

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General George S. Patton
United State Military Academy
West Point, New York

Gen. George S. Patton

distance equation 

This exercise using a map and a cursor asks the viewer to find the distance between two points.

For example, a student might be asked to find a distance from Bogota to the Panama Canal.  First the viewer would click on  Bogota and note the coordinates at the upper right.  These numbers represent pixels with an origin (0,0) at the lower left.

This is followed by clicking on the region most likely to be the Panama Canal and noting its coordinates.  The viewer then uses the distance formula, to find the measure in pixels from Bogota to the Canal. 

Finally, using these measurements and the map scale, the viewer converts the distance to miles or kilometers.  The viewer uses the scale to determine how to convert pixel distances on the map to ground measures in miles or kilometers.

The same JAVA Lite Applet map/graphic permits viewers to discover for themselves several different ways to estimate the area of an irregular region.

While both of these exercises are relatively simple mathematics, estimating is good for students of all ages. Moreover, finding areas of irregular regions is background for integral calculus.  In addition, the distance formula is derived from the Pythagorean Theorem.

The U.S. Central Intelligence Agency (CIA) provided the map for this exercise.  Should you choose to adapt this script, we suggest that you see the following link: 


Links and Books Related to Deposit #8



Dava Sobel,  Longitude: The true story of a lone genius who solved the greatest scientific problem of his time.   Fourth Estate, London, 1995. 
Greeenwich Observatory
Distances on a plane surface are simple.  Finding accurate distances, especially in sailing east and west, have been difficult in past centuries.  Much of this more complicated mathematics took place at Greenwich Observatory on the south bank of the Thames near London.
The NCB thanks Dr. Frank Wattenberg
Department of Mathematical Sciences at the United States Military Academy for Deposit #8.


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