Question

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 \(y=m x,\) 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 absolute values of the lengths of vertical feeder lines given by \(S_{2}=|4 m-1|+|5 m-6|+|10 m-3|\) Find the equation for the trunk line by this method and then determine the sum of the lengths of the feeder lines.

Short answer

The linear equation for the trunk line is \(y = mx\) where \(m\) gives the minimum \(S_2\) after substitution. The sum of the lengths of the feeder lines is the computed minimum \(S_2\) .

Step-by-step solution

Evaluate the critical points of S2

The function defined by \(S_2\) has a derivative that is undefined when the internal expression in the absolute value brackets equals 0. So, solve the equations 4m - 1 = 0, 5m - 6 = 0, and 10m - 3 = 0 to get the critical values.

Calculate derivative of S2

The derivative of \(S_2\) is piecewise because there is an absolute value in it. Divide it into three pieces according to the critical points from step 1.

Calculate the slope

Solve the derivative of \(S_2=0\). Since it is a piecewise function, remember to use the critical points obtained from Step 1 to define the correct piece to solve.

Determine the minimum sum

Substitute back the slope \(m\) obtained from the derivative into \(S_2\) to get the feeder line length. Pick the one that minimizes \(S_2\).