Determining Quilt Border Width Using A Quadratic Model
When tackling problems involving areas and dimensions, particularly when dealing with borders or frames, understanding the underlying mathematical relationships is crucial. Let's consider the question: A quilt is 5 feet wide and 6 feet long. A border around the quilt is being constructed using 25 square feet of fabric. What type of model would be appropriate to use to find the width of the border?
To solve this, we need to delve into the properties of different mathematical models and determine which one best fits the scenario. The options presented are piecewise, cubic, quadratic, and linear models. To accurately address this question, we'll dissect each model type and explore its relevance to the quilt border problem. This involves understanding how the area of the border relates to the width and length of the quilt, and how the additional area affects the overall dimensions. The solution lies in recognizing the geometric relationship created by the border and how it translates into an algebraic equation. Let's break down the problem step by step, examining the characteristics of each model to arrive at the correct answer. This exploration will not only provide the solution but also offer a deeper understanding of mathematical modeling in practical scenarios.
Dissecting the Mathematical Models
To determine the appropriate model for our quilt border problem, let's carefully examine each of the given options:
Linear Models
Linear models represent relationships with a constant rate of change, forming a straight line when graphed. They are typically expressed in the form y = mx + b, where m denotes the slope (rate of change) and b is the y-intercept (initial value). Linear models excel in scenarios involving constant growth or decline, such as the steady increase in temperature over time or the consistent decrease in the price of a commodity. However, linear models fall short when applied to situations where the rate of change is not constant. In the context of our quilt problem, a linear model would imply a direct, proportional relationship between the border width and the total area, which doesn't hold true due to the two-dimensional nature of area calculations. The area increases in a non-linear fashion as the border width increases, making a linear model an unsuitable choice for this scenario. To truly grasp the limitations of linear models, consider how the area of a square changes as its sides increase; the relationship is not linear but rather quadratic, highlighting the inadequacy of linear models for area-related problems.
Cubic Models
Cubic models involve polynomial equations of the third degree, typically written as y = ax³ + bx² + cx + d. These models are characterized by their ability to represent complex curves with two turning points, making them useful for depicting phenomena with inflection points or varying rates of change. Cubic models find applications in fields like physics, where they can describe the trajectory of a projectile under varying wind resistance, or in economics, where they can model cost functions with fixed, variable, and diminishing returns. However, cubic models are generally overkill for scenarios with simpler relationships. In our quilt border problem, the relationship between the border width and the total area is more straightforward than a cubic model can represent. The area calculation primarily involves quadratic terms due to the two-dimensional expansion, making a cubic model an unnecessarily complex choice. To illustrate, think about the volume of a cube as its sides increase; while a cubic relationship exists, the area of its faces grows quadratically, underscoring the importance of selecting the right model complexity for the given problem.
Piecewise Models
Piecewise models are defined by different functions over different intervals of their domain. They are ideal for representing situations where the relationship between variables changes abruptly at specific points. For example, a piecewise function could model the cost of electricity, which might have different rates depending on the time of day or the amount of energy consumed. These models are particularly useful in scenarios with conditional pricing, such as tiered tax systems or shipping costs that vary based on weight. However, piecewise models are not the best fit when the relationship between variables is continuous and smooth. In the context of our quilt border problem, the area added by the border changes continuously as the border width increases; there are no sudden jumps or breaks in the relationship. Therefore, a piecewise model would introduce unnecessary complexity and is not the most appropriate choice. To appreciate the utility of piecewise models, consider a scenario where a function needs to represent both a linear increase up to a certain point and then a constant value thereafter; such situations clearly demonstrate the strength of piecewise functions where a single equation would be inadequate.
Quadratic Models
Quadratic models are represented by polynomial equations of the second degree, generally expressed as y = ax² + bx + c. These models form parabolas when graphed, capturing scenarios where the rate of change is not constant but varies in a predictable manner. Quadratic models are particularly apt for describing phenomena involving areas, projectile motion, and optimization problems. In the context of our quilt border, the area added by the border increases non-linearly with the width because the border extends along both the length and the width of the quilt. This creates a relationship where the area of the border involves terms that are squared (i.e., x²), making a quadratic model the most fitting choice. The total area, including the border, can be expressed as a quadratic function of the border width, where the squared term arises from the expansion of the border around the original quilt dimensions. To further illustrate, consider the area of a circle, which is πr²; the squared term in the formula highlights the quadratic relationship between the radius and the area, reinforcing the suitability of quadratic models for area-related problems.
Applying the Quadratic Model to the Quilt Border Problem
In the case of the quilt border, the original quilt is 5 feet wide and 6 feet long, giving an area of 30 square feet. Let's denote the width of the border as x. When a border of width x is added around the quilt, the new dimensions become (5 + 2x) feet in width and (6 + 2x) feet in length. The total area of the quilt with the border is then (5 + 2x)(6 + 2x) square feet. The area of the border alone is the difference between the total area and the original quilt area, which is given as 25 square feet.
We can express this as an equation:
(5 + 2x)(6 + 2x) - 30 = 25
Expanding this equation, we get:
30 + 10x + 12x + 4x² - 30 = 25
Simplifying the equation, we have:
4x² + 22x - 25 = 0
This is a quadratic equation in the form of ax² + bx + c = 0, where a = 4, b = 22, and c = -25. The presence of the x² term confirms that a quadratic model is the appropriate choice for this problem. This equation accurately captures the relationship between the border width and the total area, reflecting the non-linear increase in area as the border width increases.
The quadratic equation can be solved using the quadratic formula or by factoring (if possible) to find the values of x, which represent the width of the border. The solutions to this equation will give us the width of the border that adds 25 square feet to the quilt's area. The quadratic nature of the relationship arises from the fact that the border increases the dimensions in two directions, both length and width, leading to a squared term in the area calculation. To further illustrate the suitability of the quadratic model, consider how the area of a picture frame is calculated; the frame adds to both dimensions, resulting in a similar quadratic relationship between the frame width and the total area. The quadratic model accurately reflects the geometric changes caused by the border, making it the perfect tool for solving this problem.
Conclusion
In summary, a quadratic model is the most appropriate for finding the width of the border around the quilt. This is because the area added by the border creates a non-linear relationship due to its extension in two dimensions, which is accurately represented by a quadratic equation. The equation 4x² + 22x - 25 = 0 effectively models the relationship between the border width and the additional area, providing a clear pathway to solving for the unknown border width. By understanding the characteristics of different mathematical models, we can effectively tackle geometric problems and gain deeper insights into the relationships between variables. Therefore, when dealing with area changes involving two-dimensional extensions, quadratic models often provide the most accurate and intuitive solutions.
