Chapter 4.7, Problem 51E

Minimum Distance In Exercises 51-53, consider a fuel distribution center located at the origin of the rectangular coordinate system (units in miles; see figures). The center supplies three factories with coordinates (4,1), (5,6), and(10, 3). A trunk line will run from the distribution center along the line $y=mx$, and feeder lines will run to the three factories. The objective is to find m such that the lengths of the feeder lines are minimized.

Minimize the sum of the squares of the lengths of the vertical feeder lines (see figure) given by

$S_1={(4m-1)}^2+{(5m-6)}^2+{(10m-3)}^2$.

Find the equation of the trunk line by this method and then determine the sum of the lengths of the feeder lines.