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

Question

?Todd's Direct, a major TV sales chain headquartered in New? Orleans, is about to open its first outlet 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.

?a) The? center-of-gravity coordinates for the location of the proposed outlet in? Mobile, Alabama should? be:

x? = ? (round your response to two decimal? places).

b) Census tracts 103 and 105 are each projected to grow by 20?% in the next year. What is the new x and y coordinate?

Census Tract Population in Census Tract X,Y Coordinates 101 2,100 (25,45) 102 5,000 (26,25) 103 10,500 (55,48) 104 7,200 (50,22) 105 10,200 (76,52) 106 20,000 (70,18) 107 12,000 (84,22)

Explanation / Answer

To be calculated:

The centre-of-gravity coordinates for the location of the proposed outlet in Mobile, Alabama

Given values:

7 census tracts, population in census tract and X,Y coordinates

Solution:

(a) The centre-of-gravity coordinates can be calculated as;

X-coordinate = Sum of [Xi x Population] / Total population in census tract

Total population in census tract = 2100 + 5000 + 10500 + 7200 + 10200 + 20000 + 12000

Total population in census tract = 67000

X-coordinate = [(25 x 2100) + (26 x 5000) + (55 x 10500) + (50 x 7200) + (76 x 10200) + (70 x 20000) + (84 x 12000)] / 67000

X-coordinate = 64.23

Y-coordinate = Sum of [Yi x Population] / Total population in census tract

Y-coordinate = [(45 x 2100) + (25 x 5000) + (48 x 10500) + (22 x 7200) + (52 x 10200) + (18 x 20000) + (22 x 12000)] / 67000

Y-coordinate = 30.39

Answer: x = 64.23, y = 30.39

(b) Census tracts 103 and 105 are projected to grow by 20% in the next year.

Census tract 103 = 10500 + (0.20 x 10500) = 12600

Census tract 105 = 10200 + (0.20 x 10200) = 12240

Total population in census tract = 2100 + 5000 + 12600 + 7200 + 12240 + 20000 + 12000

Total population in census tract = 71140

X-coordinate = [(25 x 2100) + (26 x 5000) + (55 x 12600) + (50 x 7200) + (76 x 12240) + (70 x 20000) + (84 x 12000)] / 71140

X-coordinate = 64.29

Y-coordinate = [(45 x 2100) + (25 x 5000) + (48 x 12600) + (22 x 7200) + (52 x 12240) + (18 x 20000) + (22 x 12000)] / 71140

Y-coordinate = 31.53

Answer: x = 64.29, y = 31.53

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