A farmers' cooperative association plans to build a new sugar mill in Southwestern Louisiana. The primary objective of the mill is to provide the farmers with a place to take their crop for processing that will reduce their transportation costs

The members of the co-op believe that the center-of-gravity method is appropriate for this objective. While there are over 200 sugar cane farms in the region, they are tightly clustered around six villages. Using the data below, use the center-of-gravity method to calculate the coordinates of the best location for this mill. All mileage references use the city of Lake Charles as (0,0).

Village Miles East of
Lake Charles Miles North of Lake Charles Sugar Cane tonnage
Arceneaux 90 10 240,000
Boudreaux 140 60 320,000
Cancienne 20 70 450,000
Darbonne 50 20 120,000
Evangeline 100 80 60,000
Fontenot 10 120 140,000

The center of gravity, weighted by the tonnage at each village cluster, is about 66.8 miles east of Lake Charles and 58.0 miles north of Lake Charles.

Sugar Mill Solution
Weighted # trips x-coord y-coord X multiplied Y multiplied
A 240. 90. 10. 21,600. 2,400.
B 320. 140. 60. 44,800. 19,200.
C 450. 20. 70. 9,000. 31,500.
D 120. 50. 20. 6,000. 2,400.
E 60. 100. 80. 6,000. 4,800.
F 140. 10. 120. 1,400. 16,800.
Total 1330. 410. 360. 88,800. 77,100.
Average 68.333 60.
Weighted Average 66.7669 57,9699
Median 665. 50. 60.