Find the center of mass of each two dimensional system.

Question

Four cities plan to build a new airport to serve all four communities. City B (population 180,000) is 4 mi north and
3 mi west of City A (population 75,000). City C population 240,000 is 6 mi east and 12 mi south of City A. City D
(population 105,000) is 15 mi due south of City A Find the best location for the airport.
Do not consider new roads in determining the best location.

Answer 1

From the way the problem is constructed, "best location" must mean the location that results in the lowest average travel distance, which would be the center of population, or center of mass if you treated the population numbers as "mass."

 

So start by putting each city in a 2-dimensional coordinate graph.  Since the locations of all the cities are stated relative to city A, it makes sense to put A at the origin.

 

Now, to calculate the center of mass you treat each dimension separately -- x (east-west) and y(north-south).  Calculate the "moment" for each city, by multiplying the population of the city times its x and y positions.  You would have a table:

 

       x      y      pop.      x-moment    y moment

A     0      0     75000          0                  0

B     -3     4    180000     -540000         720000

C     6    -12   240000    1440000       -2880000

D     0    -15   105000         0             -1575000

 

Now add the x and y moments.  You get (900000, -3735000)

 

Now divide the moments by the total population, which is 600000.  That gives you (1.5, -6.225).  That is the center of population in your coordinate system, which is what you were looking for -- 1.5 miles east and 6.225 miles south of city A.