Chapter 4.7, Problem 53E

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 lengths of the perpendicular feeder lines (see figure above and Exercise 77 in Section 1.2) from the trunk line to the factories given by

$S_3=\frac{|4m-1|}{\sqrt{m^2+1}}+\frac{|5m-6|}{\sqrt{m^2+1}}+\frac{|10m-3|}{\sqrt{m^2+1}}$

Find the equation of the trunk line by this method and then determine the sum of the lengths of the feeder lines. (Hint: Use a graphing utility to graph the function S₃ and approximate the required critical number.)