Problem 9 - Algebra 2

Question

For Exercises \(9-14,\) use the following information. The Future Homemakers Club is making canvas tote bags and leather tote bags for a fund-raiser. They will line both types of tote bags with canvas and use leather for the handles of both. For the canvas bags, they need 4 yards of canvas and 1 yard of leather. For the leather bags, they need 3 yards of leather and 2 yards of canvas. Their advisor purchased 56 yards of leather and 104 yards of canvas. Let c represent the number of canvas bags and let \(\ell\) represent the number of leather bags. Write a system of inequalities for the number of bags that can be made.

Step-by-Step Solution

Answer

The system of inequalities is: \(1c + 3\ell \leq 56\), \(4c + 2\ell \leq 104\), \(c \geq 0\), \(\ell \geq 0\).

Step 1: Define Variables

We are given two types of bags: canvas bags and leather bags. Let \(c\) represent the number of canvas bags and \(\ell\) represent the number of leather bags.

Step 2: Express Material Requirements

The canvas bag requires 4 yards of canvas and 1 yard of leather. The leather bag requires 2 yards of canvas and 3 yards of leather.

Step 3: Formulate Inequalities for Leather

They have 56 yards of leather available. The inequality representing the leather constraint is:\[1c + 3\ell \leq 56\]

Step 4: Formulate Inequalities for Canvas

They have 104 yards of canvas available. The inequality representing the canvas constraint is:\[4c + 2\ell \leq 104\]

Step 5: Establish Non-negativity Constraints

Since it's not possible to make a negative number of bags, the variables must satisfy:\[c \geq 0\] and \[\ell \geq 0\]

Key Concepts

Linear Programming

Linear programming is a method used to achieve the best outcome, such as maximum profit or minimum cost, in a mathematical model. This technique is crucial when deciding how to allocate resources efficiently. It uses a linear equation to find the possible solutions to a specific problem. In our exercise, linear programming helps the Future Homemakers Club determine how many canvas and leather tote bags to make based on their material constraints.

• Linear programming problems consist of variables which represent activity levels or quantities.
• It involves a linear objective function that needs to be maximized or minimized.
• Constraints in the form of linear inequalities limit the degrees of freedom available for the solution.

With clear goals and constraints, linear programming provides a structured method for solving resource allocation issues.

Inequality Constraints

Inequality constraints are an essential element in linear programming and resource allocation problems. They define the limitations or boundaries within which a solution must be found. In our scenario, inequality constraints represent the limits on the available canvas and leather.

• Constraints are usually presented in the form of inequalities, such as \(1c + 3\ell \leq 56\).
• They ensure that the solution does not exceed the available resources.
• These inequalities help maintain practical and feasible production limits.

Without these inequalities, the club could risk planning for more bags than what materials can support, leading to an ineffective solution.

Resource Allocation

Resource allocation involves distributing available resources in the most effective way to meet objectives. In the context of the exercise, this means using the available 56 yards of leather and 104 yards of canvas efficiently to produce the most tote bags.

• The first step of any resource allocation problem is assessing total resources available.
• Each resource has constraints that need to be taken into account when planning production.
• Finding a balance is crucial—producing the maximum number of bags without exceeding material limits.

Allocating resources wisely ensures that the club utilizes its materials appropriately, maximizing their output for the fundraiser.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world problem to facilitate understanding and decision making. In the tote bag exercise, mathematical modeling allows the club to plan how many bags to produce.

• It begins with identifying variables—in this case, the number of each type of bag.
• Constraints and objectives are expressed through mathematical equations and inequalities.
• The model helps visualize possible solutions and choose the optimal one.

With the use of mathematical modeling, the club can effectively forecast the relationship between their resource constraints and product outputs, leading to more informed decisions.