Problem 52 - Algebra and Trigonometry

Question

Furniture Manufacturing \(\quad\) A man and his daughter manufacture unfinished tables and chairs. Each table requires 3 hours of sawing and 1 hour of assembly. Each chair requires 2 hours of sawing and 2 hours of assembly. Between the two of them, they can put in up to 12 hours of sawing and 8 hours of assembly work each day. Find a system of inequalities that describes all possible combinations of tables and chairs that they can make daily. Graph the solution set.

Step-by-Step Solution

Answer

The system is: \( 3x + 2y \leq 12 \), \( x + 2y \leq 8 \), \( x \geq 0 \), \( y \geq 0 \).

Step 1: Define Variables

Let \( x \) represent the number of tables and \( y \) the number of chairs they manufacture daily. We need to set up inequalities based on the constraints given in the problem.

Step 2: Sawing Time Constraint

Each table requires 3 hours of sawing and each chair requires 2 hours of sawing. The total sawing time cannot exceed 12 hours daily. This gives us the inequality:\[ 3x + 2y \leq 12 \]

Step 3: Assembly Time Constraint

Each table requires 1 hour of assembly and each chair requires 2 hours of assembly. The total assembly time must not exceed 8 hours a day. This is written as:\[ x + 2y \leq 8 \]

Step 4: Non-Negative Production

Since they can't make negative numbers of tables or chairs, we have:\[ x \geq 0 \] and \[ y \geq 0 \]

Step 5: Graph the Inequalities

Plot the lines represented by the inequalities: \( 3x + 2y = 12 \) and \( x + 2y = 8 \) on a coordinate plane. Additionally, include the lines \( x = 0 \) and \( y = 0 \). Shade the region bounded by the lines and the axes in the first quadrant, which represents the feasible solution set where all constraints are satisfied.

Key Concepts

graphing inequalities

Graphing inequalities is a helpful way to visually grasp the solutions to inequality problems. For this exercise, we have two major inequalities:

• \(3x + 2y \leq 12\) - Which represents the maximum sawing hours available daily.
• \(x + 2y \leq 8\) - Which indicates the maximum assembly hours they can put in each day.

Keep in mind, we are dealing with a practical situation where negative numbers don't make sense, so \(x \geq 0\) and \(y \geq 0\). This means we focus on the first quadrant of the graph.
Start by plotting the lines of the inequalities without the inequality signs, that is, \(3x + 2y = 12\) and \(x + 2y = 8\). These lines divide the plane into regions. The points in these regions represent solutions to the inequality. You can find suitable points to plot by substituting values for \(x\) and \(y\), solving the equations.
After plotting, shade the region that satisfies all four inequalities. This shaded area shows all possible combinations of tables and chairs that fit the given constraints.

systems of inequalities

A system of inequalities is just a set of inequalities to consider at once. In this exercise, the task involves three inequalities that need to be solved together:

• The sawing time constraint: \(3x + 2y \leq 12\)
• The assembly time constraint: \(x + 2y \leq 8\)
• Non-negativity conditions: \(x \geq 0\) and \(y \geq 0\)

These inequalities define a feasible region that includes all possible solutions fitting the constraints. To solve the system, you perform a combination of algebraic manipulation and graphing.
When graphed, the solution is a region on the coordinate plane where the inequality lines intersect and which also encompasses all possible values for \(x\) and \(y\). This approach enables you to quickly see potential solutions and makes comparisons simpler. Understanding this concept is vital in optimizing resources efficiently and effectively.

constraint optimization

Constraint optimization involves finding the best solution under given restrictions. Here, the constraints are on time for sawing and assembly. The goal is to maximize the production of tables and chairs.
Once you have your feasible region from graphing the system of inequalities, you look for the best solution within this region. You can check the vertices of the polygonal region formed inside the first quadrant. Evaluating the objective function, like \(x + y\) for maximizing numbers of items produced, at these points helps determine optimal values.
If, for example, you find that producing fewer tables than chairs provides the maximum use of time and resources (and vice-versa if the opposite is true), then you utilize that mix. A practical understanding of constraint optimization can greatly enhance the decision-making process in many real-world tasks, especially in resource management.