Problem 58 - Essential Calculus

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.

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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 \(S_2\).

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions that are defined by multiple sub-functions over different intervals. They are crucial in solving optimization problems when functions change their behavior based on certain conditions. In our problem, the function \( S_2 \) is defined by a piecewise derivative because each segment is constrained by the critical points of the absolute value expressions.

• This means \( S_2 \) behaves differently across different ranges of the slope \( m \).
• Such segmentation is necessary because the absolute values in \( S_2 \) create distinct pieces where the slope changes sharply.

Understanding where one piece starts and another ends helps to break down the problem into smaller, simpler parts. By evaluating each piece separately, you can determine the overall behavior of the function.

Critical Points

Critical points are values of \(m\) where the derivative of a function is zero or undefined. They play a key role in finding both local and global minima or maxima in optimization problems. To determine these in the exercise, we solve the individual expressions inside the absolute values:

• \(4m - 1 = 0\)
• \(5m - 6 = 0\)
• \(10m - 3 = 0\)

These points are where the direction of the feeder lines might change as they represent where the length measurements inside \( S_2 \) change behavior. Examining these points helps in structuring the piecewise function properly and aids in assessing which sections to further analyze for minimization.

Absolute Value Equations

Absolute value equations are equations where the absolute function, defined as the distance from zero on a number line, is used. In our problem, \( S_2 = |4m-1| + |5m-6| + |10m-3| \) contains these to ensure all distance values are non-negative.Absolute values affect equations significantly:

• They enforce that every solution must consider the function's behavior on both sides of zero.
• Each equation transforms into two possible scenarios: one where the expression inside is positive, and another where it is negative.

We solve by considering both cases and then integrating these behaviors into the piecewise structure that defines \( S_2 \). To minimize the sum, we need to focus only on non-negative potential solutions—this is why we split \( S_2 \) into different segments based on where these changes occur.

Minimization

Minimization is the process of finding the smallest value of a function. In optimization problems, especially with absolute values and piecewise functions, it involves strategic selection of critical points and corresponding function values. In our exercise, after determining the critical points, solving \( S_2 \) involves:

• Calculating the derivative within each piece.
• Equating the derivative to zero to find the minima efficiently.
• Substituting back to find the smallest total length of the feeder line.

Understanding this involves consistent checking across all possible scenarios defined by the piecewise sections. Minimizing feeder line lengths reduces costs or resources required, effectively using such mathematical principles for practical savings.