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.

Census Tract	Population in Census Tract		​X, Y Map Coordinates
101	1,800		​(26​,46​)
102	4,800		​(25​,25​)
103	9,500		​(52​,48​)
104	6,800		​(50​,18​)
105	10,400		​(84​,48​)
106	18,000		​(70​,22​)
107	14,000		​(96​,22​)

​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).

y​ = ___ ​(round your response to two decimal​ places

​b) Census tracts 103 and 105 are each projected to grow by 30​% in the next year. The new​ center-of-gravity coordinates for the location of the proposed outlet in​ Mobile, Alabama should​ be:

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

y​ = ____ ​(round your response to two decimal​ places).

Answer 1

TRACT	LOCATION X	LOCATION Y	POPULATION
1	26	46	1800
2	25	25	4800
3	52	48	9500
4	50	18	6800
5	84	48	10400
6	70	22	18000
7	96	22	14000

1. X = SUM(X(i) * V(i)) / SUM(V(i))

(26*1800) + (25*4800) + (52*9500) + (50*6800) + (84*10400) + (70*18000) + (96*14000) / (1800 + 4800 + 9500 + 6800 + 10400 + 18000 + 14000) = 68.58

Y = SUM(Y(i) * V(i)) / SUM(Y(i))

(46*1800) + (25*4800) + (48*9500) + (18*6800) + (48*10400) + (22*18000) + (22*14000) / (1800 + 4800 + 9500 + 6800 + 10400 + 18000 + 14000) = 30.39

LOCATION = XBAR, YBAR = 68.58, 30.39

2.

TRACT	LOCATION X	LOCATION Y	POPULATION
1	26	46	1800
2	25	25	4800
3	52	48	12350
4	50	18	6800
5	84	48	13520
6	70	22	18000
7	96	22	14000

X = SUM(X(i) * V(i)) / SUM(V(i))

(26*1800) + (25*4800) + (52*12350) + (50*6800) + (84*13520) + (70*18000) + (96*14000) / (1800 + 4800 + 12350 + 6800 + 13520 + 18000 + 14000) = 68.59

Y = SUM(Y(i) * V(i)) / SUM(Y(i))

(46*1800) + (25*4800) + (48*12350) + (18*6800) + (48*13520) + (22*18000) + (22*14000) / (1800 + 4800 + 12350 + 6800 + 13520 + 18000 + 14000) = 31.86

LOCATION = XBAR, YBAR = 68.59, 31.86