Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Furniture Production A furniture company produces tables and chairs. Each table requires 2 hours in the assembly center and \(1 \frac{1}{2}\) hours in the finishing center. Each chair requires \(1 \frac{1}{2}\) hours in the assembly center and \(1 \frac{1}{2}\) hours in the finishing center. The company's assembly center is available 18 hours per day, and its finishing center is available 15 hours per day. Let \(x\) and \(y\) be the numbers of tables and chairs produced per day, respectively. (a) Find a system of inequalities describing all possible production levels, and (b) sketch the graph of the system.

Step-by-Step Solution

Answer

The system of inequalities which represent the available production levels for the company are: \(2x + 1.5y \leq 18\) and \(1.5x + 1.5y \leq 15\). And a solution to represent this system would include a graph which depicts these inequalities along with the constraints \(x \geq 0\) and \(y \geq 0\).

Step 1: Define the Variables

In this problem, let the number of tables produced by the furniture company per day be represented by \(x\) and the number of chairs produced per day be represented by \(y\).

Step 2: Create Inequalities

Given that each table costs 2 hours to assemble and each chair requires \(1 \frac{1}{2}\) hours, with the assembly center available for 18 hours per day, this gives us the inequality for assembly hours: \(2x + 1.5y \leq 18\). Similarly, the finishing hours situation translates into: \(1.5x + 1.5y \leq 15\).

Step 3: Sketch Graph

To sketch the graph, given it is a system of inequalities, each inequality should be plotted on the graph. The area of intersection of these plots will represent the feasible region, representing all possible production levels for the company.

Step 4: Don't Forget Constraints

When graphing, remember that \(x\) and \(y\) should be greater than or equal to 0, because the furniture company cannot produce a negative number of tables or chairs. This will determine the quadrant of the graph where feasible region will reside.

Key Concepts

Linear Programming

Linear programming is a method used to find the best outcome in a mathematical model whose requirements are represented by linear relationships. This technique is widely utilized in various industries for operational research, such as maximizing profits, minimizing costs, or allocating resources efficiently.

In the case of our furniture company exercise, linear programming helps determine the maximum number of tables and chairs that can be produced given the constraints on labor and resource availability. The production of tables and chairs creates a set of linear equations or inequalities representing the limiting factors, such as time constraints in the assembly and finishing centers.

The goal is to solve for 'x' (the number of tables) and 'y' (the number of chairs) that will lead to the most efficient use of resources without exceeding the labor hours available. This scenario exemplifies a linear programming problem where an optimal, or 'feasible', solution is sought.

Graphical Method

The graphical method is a visual way of solving linear programming problems, particularly when there are two variables involved. It involves sketching the constraints on a graph and identifying the feasible region where the solutions to the system of inequalities meet.

For our furniture production example, each inequality can be plotted as a line on a graph, where 'x' is the number of tables and 'y' is the number of chairs. To represent the inequality, we shade the area that satisfies the inequality. The lines often intersect, creating a polygonal area known as the feasible region.

Important Points to Remember

When sketching the graph, ensure you include:

• The correct slope and intercept for each line based on its inequality.
• Shading the appropriate side of each line according to the inequality sign.
• The feasible region where all shaded areas overlap.
• The points of intersection which can potentially represent optimal solutions.
• The axis intercepts, which sometimes represent potential optimal solutions.

This visual representation can significantly aid in identifying the possible range of solutions for 'x' and 'y'.

Feasible Region

The feasible region in a linear programming problem represented graphically is the overlapping area where all the constraints or inequalities of the problem meet. It signifies all the possible solutions that meet the given conditions.

In our furniture production scenario, any point within this region means that the combination of tables and chairs produced does not exceed the available hours in both centers. The feasible region is crucial because it visually encapsulates all the viable production plans.

Characteristics of the Feasible Region

When inspecting the feasible region, remember to note the following:

• The feasible region always appears on or between the constraint lines and never outside.
• The region is bound by the axes representing the non-negativity constraints: 'x' and 'y' cannot be negative as you can't produce negative quantities.
• Corner points of the feasible region, where the lines intersect, are of particular interest because one of these points usually corresponds to the optimal solution.

It is precisely within this enclosed space that the best or most optimal answer to our linear programming problem lies. By analyzing the feasible region, we can better understand the limitations and potential of our production capacities.