Marisol's Rectangular Frame: Defining Inequalities For Length And Width

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Introduction

In this article, we will delve into a practical problem involving Marisol, who is crafting a rectangular wooden frame. This is a classic example of how mathematical concepts, specifically systems of inequalities, can be applied to real-world scenarios. We will explore how to represent the constraints of the problem—the length of the frame and the amount of wood available—using inequalities. By understanding these inequalities, we can determine the possible dimensions of the frame that Marisol can create. This problem not only reinforces the understanding of algebraic concepts but also highlights the importance of mathematical modeling in problem-solving.

The problem presented involves translating word problems into mathematical expressions, a fundamental skill in algebra. We will break down the given information step by step, identifying the key parameters and their relationships. The length and width of the rectangular frame are the variables we need to define, and the constraints provided—the maximum length and the total amount of wood—will be expressed as inequalities. Understanding how to formulate and interpret these inequalities is crucial for solving the problem and determining the possible dimensions of the frame.

This exercise is not just about finding a numerical answer; it's about developing a logical approach to problem-solving. We will discuss how each constraint affects the possible solutions and how the system of inequalities defines a region of feasible solutions. This approach is widely applicable in various fields, such as engineering, economics, and computer science, where optimization under constraints is a common task. By the end of this article, you will have a clear understanding of how to model real-world problems using systems of inequalities and how to interpret the solutions in a practical context.

Problem Statement

Marisol is making a rectangular wooden frame. She has specific constraints to consider: the length of the frame should be no more than 12 inches, and she has less than 30 inches of wood available to use for the frame's perimeter. The challenge is to determine the system of inequalities that represents the possible length, denoted as l, and the possible width, denoted as w, of the frame.

To approach this problem, we need to translate the given information into mathematical inequalities. The first constraint states that the length of the frame should be no more than 12 inches. This can be directly translated into an inequality involving l. The second constraint involves the perimeter of the rectangle, which is determined by both the length and the width. The total amount of wood available limits the perimeter, so we need to express the perimeter in terms of l and w and then create an inequality based on the given limit of 30 inches.

This problem highlights the connection between geometry and algebra. The geometric concept of a rectangle's perimeter is combined with the algebraic concept of inequalities to model a real-world situation. Understanding this connection is crucial for applying mathematical concepts to practical problems. The solution will involve formulating two inequalities, one representing the length constraint and the other representing the perimeter constraint. These inequalities will form a system that defines the feasible region for the dimensions of the frame. Solving this system will give us a range of possible values for the length and width that satisfy Marisol's constraints.

Translating Constraints into Inequalities

To solve Marisol's problem, we must first translate the given constraints into mathematical inequalities. The constraints are:

  1. The length of the frame should be no more than 12 inches.
  2. Marisol has less than 30 inches of wood to use for the frame's perimeter.

Let's break down each constraint and express it as an inequality. The first constraint is straightforward: the length l should be no more than 12 inches. This can be written as:

l ≤ 12

This inequality states that the length l can be any value less than or equal to 12 inches. It sets an upper bound for the length of the frame. Now, let's consider the second constraint, which involves the perimeter of the rectangle. The perimeter P of a rectangle is given by the formula:

P = 2l + 2w

where l is the length and w is the width. Marisol has less than 30 inches of wood, so the perimeter of the frame must be less than 30 inches. This can be expressed as:

2l + 2w < 30

This inequality represents the constraint on the total amount of wood available. It states that the sum of twice the length and twice the width must be less than 30 inches. We can simplify this inequality by dividing both sides by 2:

l + w < 15

This simplified inequality is easier to work with and represents the same constraint on the perimeter. We now have two inequalities:

  1. l ≤ 12
  2. l + w < 15

These two inequalities form a system of inequalities that represents the constraints on the length and width of Marisol's wooden frame. In addition to these inequalities, we must also consider that the length and width cannot be negative, as they represent physical dimensions. This gives us two additional inequalities:

  1. l ≥ 0
  2. w ≥ 0

These non-negativity constraints are essential for a complete representation of the problem. They ensure that our solutions are physically meaningful. The system of inequalities now includes four inequalities, which together define the feasible region for the dimensions of the frame.

The System of Inequalities

Combining the inequalities derived from the constraints and the non-negativity conditions, we have the following system of inequalities:

  1. l ≤ 12
  2. l + w < 15
  3. l ≥ 0
  4. w ≥ 0

This system of inequalities represents all the conditions that must be satisfied for Marisol's wooden frame. The first inequality, l ≤ 12, limits the length of the frame to be no more than 12 inches. The second inequality, l + w < 15, ensures that the total perimeter of the frame is less than 30 inches (or l + w < 15 after simplification). The third and fourth inequalities, l ≥ 0 and w ≥ 0, ensure that both the length and width are non-negative, which is a physical requirement for any dimension.

This system of inequalities defines a region in the l-w plane, which represents all possible combinations of length and width that satisfy the given constraints. Any point (l, w) within this region represents a valid solution for the dimensions of the frame. The boundaries of this region are determined by the equality cases of the inequalities. For example, the line l = 12 represents the maximum possible length, and the line l + w = 15 represents the maximum possible perimeter given the available wood.

The non-negativity constraints restrict the solution to the first quadrant of the l-w plane, where both l and w are positive or zero. This is because negative dimensions are not physically meaningful. The intersection of all these inequalities forms a feasible region, which is a polygon in this case. The vertices of this polygon represent the extreme points of the solution space. Understanding the feasible region is crucial for solving optimization problems, where the goal is to find the best solution among all possible solutions.

In this context, the system of inequalities provides a complete mathematical model of the problem. It captures all the essential constraints and allows us to analyze the possible dimensions of the frame. The next step in a more complex problem might involve finding specific values for l and w that optimize some criterion, such as maximizing the area of the frame while satisfying the constraints. However, for this problem, the focus is on correctly formulating the system of inequalities that represents the given situation.

Visualizing the Solution Space

To better understand the system of inequalities, it is helpful to visualize the solution space graphically. Each inequality represents a region in the l-w plane, and the intersection of these regions represents the set of all possible solutions. Let's consider each inequality separately and then find their common region.

  1. l ≤ 12: This inequality represents the region to the left of the vertical line l = 12, including the line itself. This means that any point (l, w) with an l-coordinate less than or equal to 12 satisfies this inequality.
  2. l + w < 15: This inequality represents the region below the line l + w = 15. To graph this line, we can find two points on the line. For example, when l = 0, w = 15, and when w = 0, l = 15. The region below this line includes all points (l, w) where the sum of l and w is less than 15.
  3. l ≥ 0: This inequality represents the region to the right of the vertical line l = 0, including the line itself. This restricts the solutions to non-negative values of l.
  4. w ≥ 0: This inequality represents the region above the horizontal line w = 0, including the line itself. This restricts the solutions to non-negative values of w.

The feasible region is the intersection of all these regions. It is a polygon in the first quadrant of the l-w plane, bounded by the lines l = 0, w = 0, l = 12, and l + w = 15. The vertices of this polygon are the points where these lines intersect. These vertices are:

  • (0, 0)
  • (12, 0)
  • (0, 15)
  • (12, 3)

The feasible region includes all points inside and on the boundary of this polygon, except for the points on the line l + w = 15, as the inequality is strict (l + w < 15). Visualizing this region helps to understand the range of possible values for l and w that satisfy all the constraints. For example, the point (10, 4) lies within the feasible region, indicating that a frame with a length of 10 inches and a width of 4 inches would satisfy the given conditions.

The graphical representation of the solution space is a powerful tool for understanding and solving systems of inequalities. It provides a visual confirmation of the algebraic solution and helps in interpreting the results in the context of the problem.

Conclusion

In conclusion, we have successfully translated the constraints of Marisol's wooden frame problem into a system of inequalities. The key constraints were the maximum length of the frame (no more than 12 inches) and the limited amount of wood available for the perimeter (less than 30 inches). By expressing these constraints mathematically, we formulated the following system of inequalities:

  1. l ≤ 12
  2. l + w < 15
  3. l ≥ 0
  4. w ≥ 0

This system of inequalities represents all the conditions that must be satisfied for the dimensions of the frame. The first inequality limits the length, the second limits the perimeter, and the last two ensure that the length and width are non-negative. This mathematical model accurately captures the real-world constraints of the problem.

We also discussed the importance of visualizing the solution space graphically. By plotting the inequalities on the l-w plane, we identified the feasible region, which represents all possible combinations of length and width that satisfy the constraints. The feasible region is a polygon bounded by the lines l = 0, w = 0, l = 12, and l + w = 15. This graphical representation provides a clear understanding of the range of possible solutions and helps in interpreting the algebraic results.

This problem illustrates the power of mathematical modeling in solving real-world problems. By translating the problem into mathematical terms, we can use algebraic techniques to find solutions and gain insights. Systems of inequalities are a versatile tool for representing constraints and are widely used in various fields, such as optimization, economics, and engineering. Understanding how to formulate and solve these systems is a valuable skill for problem-solving in many contexts. The ability to break down a problem, identify the key constraints, and express them mathematically is a crucial step in the problem-solving process. This exercise demonstrates how algebra and geometry can be combined to model and solve practical problems effectively.