Problem 57 - 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 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.

Step-by-Step Solution

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 1 - 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:\[ \frac{d}{dm} S_{1} = 2(4m-1)(4) + 2(5m-6)(5) + 2(10m-3)(10) \]

Step 2 - 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.

Step 3 - 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.

Step 4 - 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 \(y = mx\).

Step 5 - 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.

Key Concepts

Differentiation

Differentiation is a powerful mathematical tool used to find the rate at which a function is changing at any point. It is crucial in optimization problems where the goal is to minimize or maximize a particular function. In the context of our exercise, we aim to minimize the sum of the squares of the feeder line lengths, which is represented by the function \(S_1=(4m-1)^2+(5m-6)^2+(10m-3)^2\).

To find where \(S_1\) is minimized, we begin by calculating its derivative with respect to \(m\). This is known as \(\frac{d}{dm} S_1\) and helps us find the rate of change of \(S_1\) with respect to \(m\):

• Use the power rule of differentiation: \((ax+b)^n\) differentiates to \(n(ax+b)^{n-1}\times a\).
• For the quadratic terms in \(S_1\), the derivative is a straightforward application of the power and constant multiplication rules.

After deriving, set the derivative equal to zero to find the stationary points. These are points where the function could potentially reach either a minimum or maximum value. Solving \(\frac{d}{dm} S_1 = 0\) gives us possible candidates for \(m\) where \(S_1\) might be minimized.

Coordinate Geometry

Coordinate geometry, or analytic geometry, merges algebra with geometry using a coordinate plane. It allows us to solve geometric problems with algebraic equations. In our problem, the fuel distribution center at the origin and the factories provide specific points on this plane: \((4,1)\), \((5,6)\), and \((10,3)\).

We utilize a line equation \(y=mx\) for the trunk line originating from the distribution center (at the origin \((0,0)\)). The slope \(m\) determines both the direction and steepness of this line. The goal is to align this trunk line's slope so that vertical lines (or feeder lines) to the three factories are minimized.

For each factory, the line connecting it to the trunk line runs vertically. Hence, understanding the geometry helps set up the equation systematically and guides us to length functions that are dependent on \(m\). For example, the vertical distance from a point \((x_i, y_i)\) to the line \(y=mx\) can be expressed algebraically as \((y_i - mx_i)\).

Quadratic Equations

Quadratic equations are mathematical statements where the highest power of the variable is squared. They take the form \(ax^2 + bx + c = 0\). In our context, when solving for \(m\) from the derivative equation \(\frac{d}{dm} S_1 = 0\), we form a quadratic equation.

To solve for \(m\), use the quadratic formula:

• \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a\), \(b\), and \(c\) are coefficients from \(am^2 + bm + c = 0\).
• Calculate the discriminant \(b^2 - 4ac\) to determine the nature of solutions.

Depending on the discriminant, the solutions can be real and distinct, real and repeated, or complex. In our exercise, real solutions exist that need to be checked. After finding potential \(m\) values, compare the results by plugging them back into the original \(S_1\) equation to confirm the one that truly minimizes the feeder lines' total length. This "real world" application of quadratic equations ensures practical and effective utility.