Problem 12 - 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. If the club plans to sell the canvas bags at a profit of \(\$ 20\) each and the leather bags at a profit of \(\$ 35\) each, write a function for the total profit on the bags.

Step-by-Step Solution

Answer

Profit function: \( P(x, y) = 20x + 35y \) with constraints \(4x + 2y \leq 104\) and \(x + 3y \leq 56\).

Step 1: Define Variables

Let's define the variables needed to formulate the function. Let \( x \) be the number of canvas tote bags, and \( y \) be the number of leather tote bags.

Step 2: Set Up System of Equations for Material Constraints

We are given that each canvas bag requires 4 yards of canvas and 1 yard of leather, while each leather bag requires 2 yards of canvas and 3 yards of leather. Therefore, we can create two equations to represent the constraints. For canvas, the equation is \( 4x + 2y \leq 104 \). For leather, the equation is \( x + 3y \leq 56 \).

Step 3: Write the Profit Function

The profit function is calculated by multiplying the number of each type of bag by their respective profits. Therefore, the total profit \( P \) as a function of \( x \) and \( y \) is given by \( P(x, y) = 20x + 35y \).

Step 4: Combine Equations and Function for Solutions

The problem is solved by considering the constraints and maximizing the profit function. Use linear programming methods like substitution or graphing to find the optimal values of \( x \) and \( y \) that will maximize \( P(x, y) = 20x + 35y \) under these constraints.

Key Concepts

Profit Maximization

Profit maximization is a central goal in many business undertakings. In our tote bag exercise, this involves setting up a function that represents how much profit can be made. Here, the Future Homemakers Club created a profit function based on two types of bags: canvas tote bags and leather tote bags.
To explain, calculate the profit earned from each type of bag and sum them up.

• The canvas bags yield a profit of \(\\(20\) each.
• The leather bags provide \(\\)35\) each.

The profit function, therefore, becomes: \[ P(x, y) = 20x + 35y \] Where:

• \(x\) is the number of canvas bags,
• \(y\) is the number of leather bags.

The goal of profit maximization is to find the combination of \(x\) and \(y\) that yields the highest total profit, under the given constraints. Finding this optimal solution requires an understanding of the limitations from the materials available and applying techniques such as linear programming.

Inequalities in Algebra

Inequalities are a fundamental part of algebra that helps express limitations or constraints on solution values. In the problem, these inequalities arise due to the materials required, given that canvas and leather resources are finite.
For example, the canvas bag uses 4 yards of canvas and 1 yard of leather. The leather bag uses 2 yards of canvas and 3 yards of leather. Therefore, we set up the following inequalities:
\[ 4x + 2y \leq 104 \] This inequality ensures that the usage of canvas does not exceed the 104 yards available. Similarly, we have:\[ x + 3y \leq 56 \] This inequality ensures that leather usage does not exceed the 56 yards available.
By incorporating these inequalities into the solution, one can graphically or algebraically determine possible values of \(x\) and \(y\) that satisfy both conditions.

System of Equations

A system of equations is essentially a collection of multiple equations that contain some of the same variables. Solving a system of equations means finding the values of these variables that satisfy all the equations simultaneously.
In our exercise, the constraints on materials (canvas and leather) lead us to two such equations:

• For canvas: \(4x + 2y \leq 104\)
• For leather: \(x + 3y \leq 56\)

These inequalities together form part of a system of constraints, integral in linear programming.
The solution can be approached through graphing methods to find feasible regions or using substitution to isolate variables. The feasible solutions must satisfy both equations under the constraints, thus aligning them with the resources available while seeking to maximize profit.