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 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.

Short answer

Calculate the derivative of \(S_{1}\) with respect to \(m\), equate it to zero and solve for \(m\). Plugging these points into \(S_{1}\), we can find which value of \(m\) minimizes it. This slope is used in the equation of the trunk line. The minimum total length of the feeder lines is found by plugging the optimal \(m\) back into the original equation \(S_{1}\).

Step-by-step solution

- Calculate the derivative of \(S_{1}\)

To optimize \(S_{1}\), we need to take its derivative with respect to \(m\), let us call it \(\frac{d}{dm} S_{1}\). Using rules of differentiation, we derive:\[ \frac{d}{dm} S_{1} = 2(4m-1)(4) + 2(5m-6)(5) + 2(10m-3)(10) \]

- Find stationary points of \(S_{1}\)

The stationary points, or the values of \(m\) where the function doesn't change, occur when the derivative equal to zero i.e. \(\frac{d}{dm} S_{1} = 0\). Using algebraic calculation, we solve for \(m\) to find these points. We could use quadratic formula or a similar technique for this.

- Check which stationary point is minimum

Solving the equation from Step 2 will possibly give us multiple solutions for \(m\). We need to verify which value of \(m\) will give us the smallest value for \(S_{1}\). We substitute these values of \(m\) into the original \(S_{1}\) equation and compare their results.

- Equation for the trunk line

Once we find the optimal slope \(m\), we can construct the equation of the trunk line which gives the minimum total length of vertical feeder lines. The equation will be \(y = mx\).

- Sum of the lengths of feeder lines

Finally, we substitute the optimal \(m\) into the original equation \(S_{1}\) to get the minimum cumulative length of the feeder lines.