Problem 53 - A Graphical Approach to College Algebra

Question

Ciolino's makes dining room furniture. A buffet requires 30 hours for construction and 10 hours for finishing, a chair 10 hours for construction and 10 hours for finishing, and a table 10 hours for construction and 30 hours for finishing. The construction department has 350 hours of labor and the finishing department has 150 hours of labor available each week. How many pieces of each type of furniture should be produced each week if the factory is to run at full capacity?

Step-by-Step Solution

Answer

Produce 10 buffets, 5 chairs, and 0 tables each week.

Step 1: Define Variables

Let's define the variables:- Let \( x \) represent the number of buffets produced each week.- Let \( y \) represent the number of chairs produced each week.- Let \( z \) represent the number of tables produced each week.

Step 2: Create Equations for Construction Hours

Each buffet requires 30 hours for construction, each chair requires 10 hours, and each table requires 10 hours. The construction department has 350 hours available. This gives us the equation:\[ 30x + 10y + 10z = 350 \]

Step 3: Create Equations for Finishing Hours

Each buffet requires 10 hours for finishing, each chair requires 10 hours, and each table requires 30 hours. The finishing department has 150 hours available. This gives us the equation:\[ 10x + 10y + 30z = 150 \]

Step 4: Simplify the Finishing Equation

The finishing equation \( 10x + 10y + 30z = 150 \) can be simplified by dividing all terms by 10, giving:\[ x + y + 3z = 15 \]

Step 5: Solve the System of Equations

We now have the system of equations:1. \[ 30x + 10y + 10z = 350 \]2. \[ x + y + 3z = 15 \]Express \( y \) in terms of \( x \) and \( z \) using the second equation: \( y = 15 - x - 3z \).Substitute \( y = 15 - x - 3z \) into the first equation:\[ 30x + 10(15 - x - 3z) + 10z = 350 \]Simplify:\[ 30x + 150 - 10x - 30z + 10z = 350 \]\[ 20x - 20z = 200 \]\[ x - z = 10 \]Substitute \( x = z + 10 \) into the simplified second equation:\[ z + 10 + y + 3z = 15 \]\[ y + 4z = 5 \]\[ y = 5 - 4z \]

Step 6: Determine Integer Solutions

Recall that \( y = 15 - x - 3z \), \( x = z + 10 \) and \( y = 5 - 4z \). Substitute \( x = z + 10 \) into \( y = 5 - 4z \) to confirm consistency with integers:\[ y = 5 - 4z \]Both expressions produce equivalent results, ensuring consistency. Now, possible integer solution occurs at \( z = 0 \):\[ x = z + 10 = 10 \]\[ y = 15 - 10 - 3(0) = 5 \]Thus, produce 10 buffets, 5 chairs, and 0 tables.

Key Concepts

System of Equations

Linear programming problems often require setting up a system of equations to solve them efficiently. In this exercise, the furniture manufacturing problem is modeled with two equations. These equations represent constraints based on the available labor hours. The first equation arises from the construction hours:

• Buffets require 30 construction hours.
• Chairs and tables each require 10 construction hours.
• The construction department has a maximum of 350 hours available.

Hence, the equation is given by:\[ 30x + 10y + 10z = 350 \]The second equation comes from the finishing hours constraint:

• Buffets require 10 finishing hours.
• Chairs also need 10 finishing hours.
• Tables require 30 finishing hours.
• 150 hours are available in the finishing department.

This equation is:\[ 10x + 10y + 30z = 150 \]Simplifying this, we use:\[ x + y + 3z = 15 \]Together, these form a system of linear equations that need solving to find how many pieces of each furniture type will utilize the labor fully.

Integer Solutions

When solving systems such as those in linear programming, especially in practical applications like manufacturing, it's crucial to find integer solutions. This is because you cannot produce a fraction of a physical item like furniture. In the given problem, we have derived expressions for each variable:

• From simplification, we have \(x = z + 10\) and \(y = 15 - x - 3z\); ensuring integer values necessitates further substitution.
• By substituting \(x\) and re-evaluating for consistent results: \(y = 5 - 4z\).

Having a feasible and useful integer solution means checking for values where all expressions align. For instance, setting \(z = 0\) yields:

• \(x = 10\)
• \(y = 5\)
• \(z = 0\)

This gives a viable integer solution: producing 10 buffets, 5 chairs, and 0 tables. The final solution must respect all original constraints and be practically implementable.

Labor Allocation

Labor allocation is about distributing available workforce hours to maximize productivity or meet specific operational targets. In linear programming, labor allocation often forms the basis for setting constraints. Here, the challenge is using the available hours in both the construction and finishing departments wisely. Each type of furniture demands a specific number of hours from each department:

• Buffet production requires a significant amount of construction time (30 hours) but less finishing time (10 hours).
• Chairs need an equal split: 10 hours for both construction and finishing.
• Tables flip the buffet's requirement, needing more finishing (30 hours) and lesser construction time (10 hours).

Meeting full capacity means assigning these hours without exceeding departmental limits while ideally using all available hours. Finding the balance allows the company to maximize its labor resources, ensuring no time is left unused. The ultimate solution respects these factors, aligning with both the mathematical solution obtained and practical business needs.