Minimum Distance In Exercises 6365, consider a fuel distribution center located at the

Chapter 4, Problem 63

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Minimum Distance

In Exercises 63–65, 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 vertical feeder lines given by

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

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

Equation Transcription:

Text Transcription:

S_1 = (4m - 1)^2 + (5m - 6)^2 + (10m  - 3)^2