Problem 46 - Precalculus: Mathematical for Calculus

Question

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. The two of them 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: \( 3x + 2y \leq 12 \), \( x + 2y \leq 8 \), \( x \geq 0 \), \( y \geq 0 \). Graph shows the feasible region.

Step 1: Define Variables

Begin by defining the variables. Let \( x \) represent the number of tables they produce per day, and \( y \) represent the number of chairs.

Step 2: Write Inequality for Sawing

Each table requires 3 hours of sawing and each chair requires 2 hours. Together, they can do at most 12 hours of sawing per day. This gives us the inequality: \[ 3x + 2y \leq 12 \]

Step 3: Write Inequality for Assembly

Each table requires 1 hour of assembly and each chair requires 2 hours. Together, they can do at most 8 hours of assembly per day. This gives us the inequality: \[ x + 2y \leq 8 \]

Step 4: Non-Negative Constraints

Since the number of tables \(x\) and chairs \(y\) cannot be negative, we have additional constraints: \[ x \geq 0 \] and \[ y \geq 0 \]

Step 5: Graph the Inequalities

Plot the lines represented by the equations derived from the inequalities on a graph: - Line for sawing time: \( 3x + 2y = 12 \)- Line for assembly time: \( x + 2y = 8 \)- Also consider \( x = 0 \) and \( y = 0 \).The feasible region where all inequalities overlap is the solution set.

Key Concepts

Graphing Inequalities

When graphing inequalities, you're dealing with regions in a coordinate system rather than single lines. The goal is to visualize which combinations of variables satisfy the given conditions.
Start by turning the inequality into an equation. For example, with the inequality \(3x + 2y \leq 12\), you would first graph \(3x + 2y = 12\). This line serves as a boundary for your inequality graph.
Choose a test point not on the boundary, often (0,0) is a convenient choice. Substitute this point into the inequality. If it satisfies the inequality, shade the region of the graph that includes the test point. If not, shade the opposite side. Repeat this process for each inequality in your system.
The intersection or overlapping areas of all shaded regions form the feasible region. This is where all conditions are met, showing the possible solutions to your problem. The boundaries themselves might be included or excluded based on their inequality symbols. For example, '<=' includes the line, while '<' does not.

Systems of Inequalities

A system of inequalities involves multiple inequalities being considered simultaneously. In real-world contexts like manufacturing, they help you understand constraints and possibilities of various actions within set limits.
For example, the exercise involves two inequalities: \(3x + 2y \leq 12\) for sawing time, and \(x + 2y \leq 8\) for assembly time. These inequalities capture the limitations on resources — specifically, limited hours available for each activity.
To solve a system graphically:

• Graph each inequality individually on the same set of axes.
• Identify the regions that satisfy each inequality.
• Intersect those regions to find where all inequalities are true simultaneously.

The intersection of all satisfying regions is the feasible region, symbolizing all the combinations that do not exceed the constraints. This is crucial in determining what combinations of tables and chairs can be produced under the given conditions.

Optimization Problems

Optimization problems are about finding the best solution or the maximum/minimum value within given constraints. They are often encountered in various fields such as production, economics, and logistics.
The exercise can be framed as an optimization problem if you aim to maximize or minimize a particular factor, like profit, within the constraints of sawing and assembly hours. For example, you might want to find the number of tables and chairs that would maximize daily revenue.
To solve optimization problems:

• Define an objective function, like total profit or cost, that you seek to optimize.
• Plot this in conjunction with your constraints (inequalities).
• Evaluate this function at the vertices or corner points of the feasible region.

The best value (maximum or minimum) will occur at one of these vertices, providing the most favorable solution under the constraints provided by the inequalities. This approach is particularly powerful when dealing with linear objectives and constraints.