Irina's Garden Fence Problem Solving Inequalities For Optimal Dimensions

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Irina faces a common challenge for gardeners: how to maximize the area of a vegetable garden while adhering to constraints. This article explores the mathematical principles behind Irina's fencing project, delving into the system of inequalities that will guide her decision-making process. We will break down the problem, explain the concepts, and demonstrate how to apply them in real-world scenarios. Let's embark on this mathematical journey to help Irina create the perfect garden!

Understanding the Problem: Setting the Stage for Irina's Garden

Irina's primary goal is to build a fence around her rectangular vegetable garden. The garden's width, a critical factor, needs to be at least 10 feet to accommodate her plants adequately. The fencing material is limited, with a maximum of 150 feet available for the perimeter. The challenge lies in determining the possible lengths (l) and widths (w) that Irina can use while staying within her constraints. This involves translating the word problem into mathematical inequalities that accurately represent the given conditions.

To fully grasp the problem, it's essential to identify the key variables and constraints. The variables are the length (l) and width (w) of the garden, both measured in feet. The constraints are:

  1. The width (w) must be at least 10 feet: This is a minimum requirement for the garden's dimensions.
  2. The total fencing available is 150 feet: This limits the perimeter of the garden.

These constraints translate into mathematical inequalities that form the system we need to solve. Let's delve deeper into formulating these inequalities.

Formulating the Inequalities: Translating Constraints into Math

To effectively model Irina's fencing problem, we need to translate the given constraints into mathematical inequalities. These inequalities will define the boundaries within which the garden's dimensions must fall. There are two primary constraints to consider:

  1. The width constraint: Irina requires the width of her garden to be at least 10 feet. Mathematically, this can be expressed as:

    w ≥ 10

    This inequality states that the width (w) must be greater than or equal to 10 feet.

  2. The perimeter constraint: The total fencing available limits the garden's perimeter to a maximum of 150 feet. The perimeter of a rectangle is calculated as:

    Perimeter = 2l + 2w

    Where 'l' is the length and 'w' is the width. Since the perimeter cannot exceed 150 feet, the inequality is:

    2l + 2w ≤ 150

    This inequality represents the limitation imposed by the available fencing material.

Combining these two inequalities, we form a system that mathematically models the possible dimensions of Irina's garden:

w ≥ 10
2l + 2w ≤ 150

This system of inequalities provides a framework for determining the acceptable lengths and widths of the garden. To fully define the feasible region, it's important to consider that length and width cannot be negative. Therefore, we also have the implicit constraints:

l ≥ 0
w ≥ 0

These additional constraints ensure that the dimensions are physically realistic. Now, let's simplify and analyze the system of inequalities to understand the possible solutions.

Simplifying the Inequalities: Making the Math Manageable

Before we can analyze the system of inequalities, it's helpful to simplify them. Simplifying the inequalities makes them easier to graph and interpret. The system of inequalities we have is:

w ≥ 10
2l + 2w ≤ 150
l ≥ 0
w ≥ 0

The second inequality, 2l + 2w ≤ 150, can be simplified by dividing both sides by 2:

l + w ≤ 75

Now, the system of inequalities becomes:

w ≥ 10
l + w ≤ 75
l ≥ 0
w ≥ 0

This simplified system is easier to work with. The inequality l + w ≤ 75 can be further rearranged to isolate 'l':

l ≤ 75 - w

This form is useful for understanding the relationship between the length and width. It shows that the length (l) must be less than or equal to 75 minus the width (w). This provides a clearer picture of how the dimensions interact within the constraints. With the inequalities simplified, we can move on to graphing them to visualize the feasible region.

Graphing the Inequalities: Visualizing the Solution Space

Graphing the inequalities is a powerful method for visualizing the possible solutions for Irina's garden dimensions. Each inequality represents a region on the coordinate plane, and the intersection of these regions forms the feasible region, which contains all the valid combinations of length (l) and width (w).

To graph the inequalities, we'll treat 'l' as the x-axis and 'w' as the y-axis. Let's graph each inequality step by step:

  1. w ≥ 10: This inequality represents a horizontal line at w = 10. Since it's a 'greater than or equal to' inequality, we draw a solid line and shade the region above the line, indicating that all points with a y-coordinate (width) of 10 or greater are valid.

  2. l + w ≤ 75: First, we graph the line l + w = 75. To do this, we can find two points on the line. For example, if l = 0, then w = 75, and if w = 0, then l = 75. Plotting these points (0, 75) and (75, 0) and drawing a line through them gives us the boundary. Since it's a 'less than or equal to' inequality, we draw a solid line and shade the region below the line.

  3. l ≥ 0: This inequality represents a vertical line at l = 0 (the y-axis). We draw a solid line and shade the region to the right, indicating that all points with a positive x-coordinate (length) are valid.

  4. w ≥ 0: This inequality represents a horizontal line at w = 0 (the x-axis). We draw a solid line and shade the region above, indicating that all points with a positive y-coordinate (width) are valid.

The feasible region is the area where all shaded regions overlap. This area represents all possible combinations of length and width that satisfy the given constraints. The vertices of this feasible region are particularly important because they represent the extreme points of the solution space.

Identifying the Feasible Region: Finding the Overlap

Identifying the feasible region involves finding the area on the graph where all the inequalities are simultaneously satisfied. This region represents all possible combinations of length (l) and width (w) that Irina can use for her garden while adhering to her constraints.

After graphing the inequalities, the feasible region is the polygon formed by the intersection of the shaded areas. It's bounded by the lines representing the inequalities w ≥ 10, l + w ≤ 75, l ≥ 0, and w ≥ 0. The vertices of this polygon are the points where the boundary lines intersect. These vertices are crucial because they represent the extreme values of the feasible region and can help Irina optimize her garden's dimensions.

To find the coordinates of the vertices, we need to solve the systems of equations formed by the intersecting lines:

  1. Intersection of w = 10 and l = 0: This point is (0, 10).
  2. Intersection of w = 10 and l + w = 75: Substituting w = 10 into the equation l + w = 75, we get l + 10 = 75, so l = 65. This point is (65, 10).
  3. Intersection of l + w = 75 and l = 0 is not a vertex because of the constraint w>=10.
  4. Intersection of l + w = 75 and w = 0 is not a vertex because of the constraint w>=10.

The vertices of the feasible region are therefore (0,10) and (65,10). These points represent the extreme combinations of length and width that satisfy the constraints. Now, let's interpret these points and understand their implications for Irina's garden design.

Interpreting the Results: Maximizing the Garden's Potential

Interpreting the results of the system of inequalities allows Irina to make informed decisions about the dimensions of her garden. Interpreting these results means understanding what the feasible region and its vertices tell us about the possible lengths and widths that Irina can use.

The feasible region represents all valid combinations of length (l) and width (w) that satisfy the constraints: a minimum width of 10 feet and a maximum perimeter of 150 feet. The vertices of the feasible region, which we found to be (0, 10) and (65, 10), are particularly important. The point (0, 10) indicates the smallest possible length with the width requirement and (65, 10) indicates the largest length with the width requirement.

The solution (65, 10) tells Irina that, if she chooses the minimum width of 10 feet, she can maximize the length to 65 feet while still staying within the 150-foot fencing limit. This combination would give her the largest possible area for her garden within the given constraints. She might choose a length shorter than 65 feet, but that would reduce the overall area of the garden.

By understanding the feasible region and its vertices, Irina can make an informed decision about the dimensions of her garden, balancing her desire for space with the limitations of her fencing material. This mathematical analysis provides a solid foundation for her garden planning.

In conclusion, by formulating and simplifying inequalities, graphing them, identifying the feasible region, and interpreting the results, we've provided Irina with a comprehensive understanding of her fencing options. This approach not only solves the immediate problem but also illustrates the power of mathematical modeling in real-world decision-making. Irina can now confidently plan her vegetable garden, knowing she has optimized the dimensions to make the most of her resources. This example highlights how mathematical principles can be applied to everyday situations to achieve the best possible outcome.

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What are the system of inequalities that models the possible lengths, l, and widths, w, of Irina's rectangular vegetable garden, given that it has a width of at least 10 feet and she can use a maximum of 150 feet of fencing?

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Irina's Garden Fence Problem Solving Inequalities for Optimal Dimensions