Problem 10 - 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. Draw the graph showing the feasible region.

Step-by-Step Solution

Answer

The feasible region is the area on the graph that satisfies all the inequalities simultaneously: \( 4x + 2y \le 104 \) and \( x + 3y \le 56 \), \( x \geq 0 \), \( y \geq 0 \).

Step 1: Define Variables and Constraints

Let's define two variables: \( x \) for the number of canvas tote bags and \( y \) for the number of leather tote bags. For canvas bags, we need 4 yards of canvas and 1 yard of leather. For leather bags, we need 3 yards of leather and 2 yards of canvas. We have total constraints of 56 yards of leather and 104 yards of canvas purchased, leading to the following system of inequalities:1. \( 4x + 2y \leq 104 \) (canvas constraint)2. \( x + 3y \leq 56 \) (leather constraint)3. \( x \geq 0 \) and \( y \geq 0 \) (non-negative bags).

Step 2: Graph the Inequalities on a Coordinate Plane

To graph the inequalities, start by considering the equations of the lines: - For \( 4x + 2y = 104 \), rearranging gives \( y = 52 - 2x \). The intercepts are (0, 52) and (26, 0).- For \( x + 3y = 56 \), rearranging gives \( y = \frac{56 - x}{3} \). The intercepts are (0, \(\frac{56}{3}\)) and (56, 0).Plot these lines on a coordinate plane. Shade the region below these lines as they represent the inequality \( \leq \). Ensure to only consider the first quadrant since \( x \geq 0 \) and \( y \geq 0 \).

Step 3: Identify the Feasible Region

The feasible region is the area on the graph where all shaded regions from the inequalities overlap. It is bounded by parts of the lines \( 4x + 2y = 104 \), \( x + 3y = 56 \), the x-axis, and the y-axis. This area represents all combinations of \( x \) and \( y \) that satisfy the constraints simultaneously.

Key Concepts

Inequalities

In the context of linear programming, inequalities are used to model constraints that a problem must satisfy. An inequality establishes a range of permissible values within which a solution must fall for the problem to be viable.

For our exercise involving the production of tote bags, inequalities are derived from the available material constraints. Each inequality corresponds to a specific resource limitation:

• Canvas usage: The inequality \(4x + 2y \leq 104\) ensures that the total canvas used for making canvas bags (\(x\)) and leather bags (\(y\)) does not exceed 104 yards.
• Leather usage: The inequality \(x + 3y \leq 56\) ensures that the total leather used does not exceed 56 yards.

These inequalities guide us to find feasible solutions within the available resources. They are vital for visualizing restricted regions on a graph that represent conditions each possible solution must meet.

Feasible Region

In linear programming, the feasible region is the set of points that satisfies all inequalities simultaneously. It is crucial because this region encompasses all possible solutions that meet the problem's constraints.

When we plot the given inequalities on a coordinate plane, the feasible region appears as the overlapping shaded area that meets all conditions laid out by the inequalities. It is bounded by:

• The lines representing each constraint (e.g., \(4x + 2y = 104\) and \(x + 3y = 56\)).
• The non-negativity constraints which restrict solutions to the first quadrant, due to \(x \geq 0\) and \(y \geq 0\).

Within this feasible region, any point represents a possible and acceptable solution—meaning it adheres to all specified constraints. For our problem, the feasible region will show all the potential combinations of canvas and leather bags that can be produced without exceeding materials in stock.

Coordinate Plane

A coordinate plane is a two-dimensional surface formed by two intersecting lines: the x-axis and the y-axis. Each point on this plane is identified by an ordered pair \((x, y)\).

The coordinate plane plays a vital role in visualizing linear inequalities and their solutions. In our tote bag problem, we use the coordinate plane to plot the lines derived from each inequality constraint. This helps in determining the feasible region where solutions exist.
Steps to using the coordinate plane in this context include:

• Rearranging each inequality into its line equation form (e.g., \(y = 52 - 2x\)).
• Plotting these lines on the graph by finding x and y intercepts.
• Shading the appropriate regions according to inequality signs \((\leq)\).

This visual method ensures that the graphical solution aligns with the algebraic one, offering a clearer understanding of where viable solutions are located among the graph lines.

Systems of Equations

A system of equations is comprised of multiple equations that are solved together because they share common variables. In the context of linear programming, these systems define the constraints under which the solution must be found.

For the exercise at hand, our system of equations includes:

• \(4x + 2y \leq 104\), representing the canvas constraint.
• \(x + 3y \leq 56\), representing the leather constraint.

By solving this system graphically, we find the feasible region. The solution must be a set of values for \(x\) and \(y\) that satisfy all equations, usually represented by the intersection of the corresponding inequalities plotted on a coordinate plane.

Understanding these systems is key to interpreting the resources and limitations within the problem, ensuring that the final decision or solution is both optimal and practical.