Question

Consider the three-floor building in which there are eight offices. The manager of the building wants to find a suitable location for a mail distribution room used collectively by all the offices. The existing offices are located at the points P₁=(15,25,−8),P₂=(40,20,2), P₃=(8,20,2),P₄=(30,40,2),P₅=(25,35,12),P₆=(25,35,12),P₇=(25,35,22), and P₈=(12,10,22). Office 1 is on the ground floor, Office 2, Office 3 , and Office 4 are on the 1ˢᵗ floor, Office 5 and Office 6 are on the 2ⁿᵈ floor, and finally, Office 7 and Office 8 are on the 3ᵣ floor.

The weights for the existing offices are 1,2,2,3,1,2,2, and 1 . Assuming rectilinear distance is a reasonable approximation of the actual travel distance, what is the location of the mail distribution room which will minimize the total distance that the people working in these offices have to travel?

Answer 1

The mail distribution room should be located at the centroid of the offices to minimize the total distance traveled by the employees.

Step-by-step explanation:

To minimize the total distance that the people working in these offices have to travel, the mail distribution room should be located at the centroid of the offices. The centroid is the average of the coordinates of the offices, weighted by the number of people working in each office. The coordinates of the centroid can be calculated using the formula:

x-coordinate of centroid = (Σ(xi * wi)) / (Σwi)

y-coordinate of centroid = (Σ(yi * wi)) / (Σwi)

z-coordinate of centroid = (Σ(zi * wi)) / (Σwi)

Using the given values, we can calculate the coordinates of the centroid to be approximately (21.063, 26.625, 2.125).