Problem 23 - College Algebra

Question

In \(1978,\) a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon \(20 .\) To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{lcc}\hline & \text { Boeing 727 } & \text { Falcon 20 } \\\\\hline \text { Direct Operating cost } & \$ 1400 \text { per hour } & \$ 500 \text { per hour } \\\\\text { Payload } & 42,000 \text { pounds } & 6000 \text { pounds }\end{array}$$ Federal Express was faced with the following constraints: \(\cdot\) Hourly operating cost was limited to \(\$ 35,000 .\) \(\cdot\) Total payload had to be at least \(672,000\) pounds. \(\cdot\) Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?

Step-by-Step Solution

Answer

The solution will be the maximum number of each aircraft they can purchase without violating the given conditions as achieved from linear programming optimization

Step 1: Problem Formulation

Let \(x\) be the number of Boeing 727s and \(y\) be the number of Falcon 20s. The constraints are as follows, the cost was limited to \$35,000, meaning \(1400x + 500y <= 35000\). The payload had to be at least 672,000 pounds which is \(42000x + 6000y >= 672000\). Also, the number of 727s were limited to 20, hence \(x <= 20\). The problem is to maximize the number of aircrafts that is \(x + y\)

Step 2: Graphical solution

To find the solution, it's best to sketch the inequalities on a graph and find the feasible region. Where all constraints overlap is the feasible region. Then, inspect the boundary of the feasible region to find the vertex points. The vertex points give possible values for \(x\) and \(y\)

Step 3: Determine optimal solution

The optimal solution corresponds to the vertex of the feasible region that provides the greatest values for \(x + y\). Thorough inspection of these values will provide the solution

Key Concepts

Constraints

Constraints are limitations or restrictions that must be taken into account when solving a linear programming problem. In the given exercise, Federal Express was bound by three specific constraints when deciding on the number and type of aircraft to purchase.

One constraint relates to the **hourly operating cost**, set at a maximum of $35,000. This means that the combined hourly costs from operating the Boeing 727s and Falcon 20s should not exceed this amount. The equation representing this constraint is written as:

• \(1400x + 500y \leq 35000\)

Next is the **payload constraint**, requiring a total payload of at least 672,000 pounds. Essentially, the weight capacity provided by all purchased aircraft must meet or surpass this threshold:

• \(42000x + 6000y \geq 672000\)

Moreover, they could only acquire a **limited number of Boeing 727s**, specifically a maximum of 20 aircraft:

• \(x \leq 20\)

Understanding these constraints is crucial as they affect the decision-making process directly, ensuring the solutions are feasible under real-world conditions.

Graphical Solution

The graphical solution method provides a visual way to identify feasible solutions for linear programming problems. By graphing the inequalities on a coordinate plane, we can identify the region that satisfies all constraints simultaneously.

Here's a simple approach to achieve this:

• First, convert each inequality into an equation to graph the boundary lines of the constraints.
• Plot these lines on a coordinate graph, where the x-axis denotes the number of Boeing 727s, and the y-axis represents the Falcon 20s.
• The area where all shaded regions overlap is known as the **feasible region**. This region contains all possible solutions that satisfy the given constraints.

In the feasible region, specific points known as **vertices** or corner points hold particular importance. These points represent potential optimal solutions that can be evaluated further. By calculating the values of the objective function, \(x + y\), at each vertex, we can determine which combination of aircraft maximizes their number. This method is inherently visual, helping to simplify and solidify the understanding of feasible solutions.

Inequalities

In linear programming, inequalities are used to express constraints and conditions in a mathematical form. They establish relationships between variables that need to be satisfied to find viable solutions. Understanding these inequalities is key to solving problems like the one faced by Federal Express.

Here are the primary types of inequalities used:

• **Linear inequalities** form the backbone of constraints, conveying restrictions like budget limits or capacity requirements. For example, the inequality \(1400x + 500y \leq 35000\) assures that operating costs do not exceed allowed limits.
• **Inequality signs** (">=", "<=") represent conditions that must be fulfilled. "<=" denotes an upper limit scenario, while ">=" defines a minimum requirement.
• In our exercise, each inequality specifies a criterion limiting the solution set, ensuring it only includes combinations of Boeing 727s and Falcon 20s that meet every condition simultaneously.

By graphing these inequalities on the coordinate plane, the feasible area is revealed as a visual representation of all solutions that fulfill the required conditions. Mastery of using inequalities and translating them graphically provides a robust approach to tackling linear programming challenges effectively.