This is a weighted center (center of mass) problem, where populations are the weights. Since we are not considering new roads, the best airport location minimizes total travel distance and is found by the population-weighted average of coordinates.
Step 1: Set up coordinates
City | Coordinates (x, y) | Population
A | (0, 0) | 75,000
B | (−3, 4) | 180,000
C | (6, −12) | 240,000
D | (0, −15) | 105,000
Step 2: Compute weighted averages
Total population:
75,000 + 180,000 + 240,000 + 105,000 = 600,000
x-coordinate:
[(75,000)(0) + (180,000)(-3) + (240,000)(6) + (105,000)(0)] / 600,000 = 1.5
y-coordinate:
[(75,000)(0) + (180,000)(4) + (240,000)(-12) + (105,000)(-15)] / 600,000 = -6.225
Therefore, the best airport location would be at the point (1.5, -6.225) with respect to City A.
This location balances travel distance according to population size and gives greater influence to the larger cities (especially City C).
Thus, our final answer is 1.5 miles east and 6.2 miles south of City A.
