Todd\'s Direct, a major TV sales chain headquartered in New Orleans, is about to

Question

Todd's Direct, a major TV sales chain headquartered in New Orleans, is about to open its first oulet in Mobile, Alabama, and wants to select a site that will place the new outlet in the center of Mobile's population base. Todd examines the seven census tracts in Mobile, plots the coordinates of the center of each from a map, and looks up the population base in each to use as a weighting. The information gathered appears in the following table. X, Y Map Population in Census Tract Census Tract 101 102 103 104 105 106 107 2,100 4,800 9,500 7,000 10,200 20,000 12,000 Coordinates (25,46) (24,24) (55,48) (50,18) (76,50) (68,18) (90,22) a) The center-of-gravity coordinates for the location of the proposed outlet in Mobile, Alabama should be

Explanation / Answer

CG

X co ordinates = sum of (Xi Wi) ( i=1 to n) / sum of Wi ( i= 1 to n)

= 2100x25+ 4800x24+9500x55+7000x50+10200x76+20000x68+12000x90 / 2100+...12000

= 52500+115200+522500+350000+775200+1360000+1080000 / 65600

= 64.86

Y co ordinates = 2100x46+4800x24+9500x48+7000x18+10200x50+20000x18+12000x22 / 65600

=96600+115200+456000+126000+510000+360000+264000 /65600

=29.38

If the tracts 103 qnd 105 grow by 20% each, the new values of population will increase by 1900 and 2040 respectively.

New value of CG =

Fraction increase in X = 1900x55+2040x76 / 65600+3940

= 104500+155040 /69540 =3.73

New X coordiante = 64.86+3.73 =68.59

Fraction increase in Y = 1900x48+2040x50 / 69540 =91200+102000 / 69540 = 2.778

New Y co ordinate = 32.15

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