Problem 49 - Elementary Algebra Within Reach

Question

A furniture company can sell all the tables and chairs it produces. Each table requires 1 hour in the assembly center and \(1 \frac{1}{2}\) hours in the finishing center. Each chair requires 2 hours in the assembly center and \(\frac{3}{4}\) hour in the finishing center. The company's assembly center is available 12 hours per day, and its finishing center is available 18 hours per day. Write a system of linear inequalities that describes the different production levels. Graph the system.

Step-by-Step Solution

Answer

The system of inequalities that describe the different production levels are: \( T + 2C \leq 12 \), \( \frac{3}{2}T + \frac{3}{4}C \leq 18 \), \( T \geq 0 \), \( C \geq 0 \)

Step 1: Define the variables

Assign variables to the amount of each product produced, for example, let \(T\) represent the number of tables and \(C\) represent the number of chairs.

Step 2: Create inequalities for assembly time

Each table requires 1 hour of assembly time and each chair requires 2 hours. So the total assembly time in a day which is 12 hours, gives the inequality: \( T + 2C \leq 12 \)

Step 3: Create inequalities for finishing time

For the finishing time for tables and chairs, (each table requires \(1 \frac{1}{2}\) hours and each chair requires \(\frac{3}{4}\) hour), respecting the available 18 hours per day, we formulate a second inequality: \( \frac{3}{2}T + \frac{3}{4}C \leq 18 \)

Step 4: Non-negativity condition

Since we cannot make a negative number of tables or chairs, we also have the constraints: \( T \geq 0 \) and \( C \geq 0 \)

Step 5: Graph the system of inequalities

Plot these inequalities on a graph. The solution will be the area of the graph where all the given conditions are satisfied

Key Concepts

Systems of Equations

In the world of mathematics, a system of equations is a set of two or more equations with the same variables. These equations allow us to find values for those variables that satisfy all the equations in the system simultaneously.

• In our example, each equation represents a constraint for the furniture company based on available resources, like time in the assembly and finishing centers.
• The company needs to determine how many tables (T) and chairs (C) they can produce per day without exceeding available assembly and finishing hours.
• To do this, we define two key inequalities: one for assembly time (\( T + 2C \leq 12 \)) and another for finishing time (\( \frac{3}{2}T + \frac{3}{4}C \leq 18 \)).

When dealing with systems of inequalities, we're seeking an overlap or "solution region," where all constraints are satisfied. This approach ensures that all production activities remain within resource limits, guiding efficient production planning.

Graphing Inequalities

Graphing inequalities is a crucial step in visualizing the constraints imposed by a system of equations. It helps us identify the feasible region where all conditions are met.

• When graphing each inequality, it's essential to plot the boundary line by treating the inequality as an equation. For example, for \( T + 2C = 12 \), plot the line where all points satisfy this equation.
• Next, determine which side of the line satisfies the inequality. Select a test point, such as the origin (0,0), and substitute into the inequality. If true, shade that side.
• Repeat the process for the second inequality, \( \frac{3}{2}T + \frac{3}{4}C \leq 18 \), and ensure that the shading is in the direction that satisfies the inequality.

The overlapping shaded area of all inequalities in the same graph represents possible solutions. This region indicates all combinations of tables and chairs the company can produce without overextending their resources.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-life situations using mathematical expressions and structures. This technique helps in decision making by providing visual and quantitative analysis.

• For the furniture company, mathematical modeling simplifies complex production constraints into comprehensible inequalities and visual graphs.
• By converting resource limits like assembly and finishing hours into inequalities, the company gains a clearer understanding of production capabilities.
• This model helps in foreseeing possible issues, such as resource shortages or excesses, allowing the company to adjust their strategy accordingly.

Using mathematical models, companies can optimize their operations while staying within predefined constraints, leading to more efficient and effective business practices.