Problem 46 - Algebra and Trigonometry

Question

Furniture Manufacturing 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 inequalities are \( 3x + 2y \leq 12 \), \( x + 2y \leq 8 \), \( x \geq 0 \), \( y \geq 0 \). Graph the region they define.

Step 1: Define Variables

Let \( x \) be the number of tables manufactured per day, and \( y \) be the number of chairs manufactured per day. Our goal is to determine the possible combinations of \( x \) and \( y \) given the constraints on sawing and assembly work.

Step 2: Set Up Inequality for Sawing

Each table requires 3 hours of sawing and each chair requires 2 hours of sawing. Since they can spend at most 12 hours in total on sawing per day, the inequality is \( 3x + 2y \leq 12 \).

Step 3: Set Up Inequality for Assembly

Each table requires 1 hour of assembly and each chair requires 2 hours of assembly. Since they can spend at most 8 hours in total on assembly per day, the inequality is \( x + 2y \leq 8 \).

Step 4: Non-negativity Constraints

Since it is not possible to manufacture a negative number of tables or chairs, we also have the constraints \( x \geq 0 \) and \( y \geq 0 \).

Step 5: Graph the System of Inequalities

To graph the system of inequalities, first draw the lines corresponding to the equalities: \( 3x + 2y = 12 \) and \( x + 2y = 8 \). Then shade the region that satisfies both inequalities, including the first quadrant where both \( x \) and \( y \) are non-negative. The feasible region will be a polygonal region on the graph.

Key Concepts

Systems of Inequalities

A system of inequalities is similar to a system of equations, except instead of equalities, it involves inequalities. In real-world scenarios, like the furniture manufacturing exercise, systems of inequalities are crucial for representing constraints. In this scenario:

• Sawing time is limited, giving us the inequality \(3x + 2y \leq 12\).
• Assembly time is restricted, leading to \(x + 2y \leq 8\).
• Non-negativity of output ensures \(x \geq 0\) and \(y \geq 0\).

These inequalities collectively form a system that describes the boundaries within which the manufacturers must operate. Solving a system of inequalities typically involves finding a range of solutions, each satisfying all the constraints imposed by the inequalities.

Graphing Inequalities

Graphing inequalities is a method to visually represent the solutions of each inequality in a system. Here's a straightforward approach to handling it:

• Start by graphing each inequality as if it were an equation. This turns it into a line on the coordinate plane.
• For example, the lines \(3x + 2y = 12\) and \(x + 2y = 8\) divide the plane into regions.
• Next, determine which side of each line satisfies the inequality, using a test point (usually the origin, if it's not on the line).
• Shade the region that fulfills the inequality requirements for each constraint. Only the overlapping shaded area fulfills all inequalities concurrently.

This graphical approach makes it easy to visualize the limits within which you can work within the system, making sure that all given conditions are satisfied.

Feasible Region

In systems of inequalities, the feasible region is the set of all possible solutions to the system. It is typically represented graphically as the area where all of the inequalities in the system overlap.

• After shading the appropriate regions in the graph, the feasible region will be the intersection area where all shaded areas overlap.
• For the furniture manufacturing problem, this region is defined by the combination of sawing and assembly constraints, taking into account non-negative values of \(x\) and \(y\).
• Within the feasible region, any point represents a valid and possible combination of tables and chairs the manufacturers can produce without exceeding their resource constraints.

Identifying the feasible region allows decision-makers to optimize production within constraints, paving the way for efficient allocation and resource management.