Marisol's Wooden Frame System Of Inequalities For Length And Width
Marisol is embarking on a woodworking project, aiming to construct a rectangular wooden frame. She has some constraints to work with: the length of the frame should not exceed 12 inches, and she has a limited supply of wood, less than 30 inches. The challenge lies in determining the possible dimensions, specifically the length and width, of the frame while adhering to these limitations. This problem can be elegantly solved using a system of inequalities, a powerful tool in mathematics for representing constraints and finding feasible solutions.
Understanding the Problem: Length, Width, and Perimeter
To effectively model this situation with inequalities, it's crucial to define the variables and translate the given information into mathematical expressions. Let's denote the length of the frame as "l" and the width as "w," both measured in inches. The first constraint states that the length should be no more than 12 inches. This can be directly translated into the inequality l ≤ 12. This inequality sets an upper bound for the length, ensuring it doesn't exceed the specified limit.
The second constraint involves the amount of wood available. The total length of wood required to construct the frame corresponds to the perimeter of the rectangle. The perimeter of a rectangle is given by the formula P = 2l + 2w. Marisol has less than 30 inches of wood, which translates to the inequality 2l + 2w < 30. This inequality represents the limitation on the total perimeter, ensuring it remains within the available wood supply.
These two inequalities, l ≤ 12 and 2l + 2w < 30, form a system of inequalities that mathematically describes the constraints of the problem. Solving this system will reveal the possible combinations of length and width that satisfy both conditions.
Setting up the System of Inequalities
Now, let's formally define the system of inequalities that represents Marisol's wooden frame project. We have already identified the two key inequalities:
- l ≤ 12: This inequality restricts the length of the frame to be no more than 12 inches.
- 2l + 2w < 30: This inequality limits the total perimeter of the frame to be less than 30 inches, reflecting the limited wood supply.
These two inequalities, taken together, constitute the system of inequalities that we need to solve. However, there's an implicit constraint that we haven't explicitly stated yet: both the length and width of the frame must be non-negative. This is a practical consideration, as we cannot have a frame with a negative length or width. This gives us two additional inequalities:
- l ≥ 0: The length of the frame must be greater than or equal to zero.
- w ≥ 0: The width of the frame must be greater than or equal to zero.
Therefore, the complete system of inequalities that represents the possible dimensions of Marisol's wooden frame is:
- l ≤ 12
- 2l + 2w < 30
- l ≥ 0
- w ≥ 0
This system of inequalities provides a comprehensive mathematical representation of the problem, capturing all the relevant constraints. The next step involves solving this system to determine the feasible region, which represents all possible combinations of length and width that satisfy the conditions.
Solving the System of Inequalities: Graphically Representing the Solution
The most intuitive way to solve a system of inequalities is by graphically representing each inequality on a coordinate plane. The region where all the inequalities overlap represents the solution set, also known as the feasible region. In this case, the x-axis will represent the length (l), and the y-axis will represent the width (w). Let's graph each inequality step by step.
Graphing l ≤ 12
The inequality l ≤ 12 represents all points where the length is less than or equal to 12. To graph this, we first draw a vertical line at l = 12. Since the inequality includes "less than or equal to," the line is solid, indicating that the points on the line are also part of the solution. We then shade the region to the left of the line, representing all lengths less than 12.
Graphing 2l + 2w < 30
To graph the inequality 2l + 2w < 30, it's helpful to rewrite it in slope-intercept form (w < ml + b). Dividing both sides by 2, we get l + w < 15. Then, subtracting l from both sides, we have w < -l + 15. This represents a line with a slope of -1 and a y-intercept of 15. Since the inequality is "less than," we draw a dashed line to indicate that the points on the line are not included in the solution. We then shade the region below the line, representing all widths less than -l + 15.
Graphing l ≥ 0 and w ≥ 0
The inequalities l ≥ 0 and w ≥ 0 restrict the solution to the first quadrant of the coordinate plane. l ≥ 0 represents all points to the right of the y-axis (including the y-axis), and w ≥ 0 represents all points above the x-axis (including the x-axis).
Identifying the Feasible Region
The feasible region is the area where all shaded regions overlap. In this case, it's a quadrilateral bounded by the lines l = 0, w = 0, l = 12, and w = -l + 15. Any point within this region represents a possible combination of length and width that satisfies all the constraints of Marisol's wooden frame project. The coordinates of the points within this region will provide the possible dimensions for the frame, ensuring that the length does not exceed 12 inches and the total perimeter remains less than 30 inches.
Interpreting the Solution: Possible Dimensions for the Frame
The feasible region, as determined by the graphical solution, provides a visual representation of all possible lengths and widths for Marisol's wooden frame. Any point (l, w) within this region satisfies the conditions l ≤ 12 and 2l + 2w < 30, along with the non-negativity constraints l ≥ 0 and w ≥ 0. To interpret the solution, we can consider some specific points within the feasible region.
Example Points
- (l = 5, w = 5): This point lies well within the feasible region. A frame with a length of 5 inches and a width of 5 inches satisfies both constraints. The perimeter would be 2(5) + 2(5) = 20 inches, which is less than the 30-inch limit.
- (l = 10, w = 2): This point also falls within the feasible region. A frame with a length of 10 inches and a width of 2 inches is a valid solution. The perimeter would be 2(10) + 2(2) = 24 inches, which is within the limit.
- (l = 12, w = 1): This point lies on the boundary l = 12 and within the feasible region. A frame with a length of 12 inches and a width of 1 inch is acceptable. The perimeter would be 2(12) + 2(1) = 26 inches, still less than 30 inches.
- (l = 12, w = 2): This point, however, lies outside the feasible region. While the length is at the maximum allowed value of 12 inches, the perimeter would be 2(12) + 2(2) = 28 inches, and it is less than the limit wood supply of 30 inches. So, this point can be considered a solution.
Practical Considerations
While the feasible region provides a range of mathematical solutions, practical considerations might further influence Marisol's choice of dimensions. For instance, she might want a frame that is more aesthetically pleasing, which could lead her to choose a length and width that are closer in value. Or, she might have a specific picture or artwork in mind that she wants to frame, which would dictate the dimensions of the frame. The system of inequalities provides a solid foundation for decision-making, but ultimately, Marisol's artistic vision and practical needs will guide her final choice.
Conclusion: Systems of Inequalities as a Problem-Solving Tool
Marisol's wooden frame project beautifully illustrates the power of systems of inequalities in solving real-world problems. By translating the given constraints into mathematical expressions, we were able to create a system of inequalities that accurately represents the problem. The graphical solution provided a clear visualization of the feasible region, allowing us to identify all possible combinations of length and width that satisfy the conditions. This approach not only provides a solution but also offers a range of possibilities, enabling informed decision-making.
Systems of inequalities are not limited to woodworking projects; they have wide-ranging applications in various fields, including economics, engineering, and computer science. They are a valuable tool for modeling constraints, optimizing solutions, and making informed decisions in complex situations. Understanding and applying systems of inequalities empowers us to tackle real-world problems effectively and efficiently.
