Solving For The Original Side Length Of Stacey's Cloth Square
Introduction: The Cloth Cutting Puzzle
In this intriguing mathematical puzzle, we delve into the world of geometry and area calculations. The problem revolves around Stacey's square piece of cloth, a seemingly simple object that holds a fascinating secret. She makes a precise modification – cutting 3 inches off both the length and the width – transforming her original square into a smaller one. The crux of the problem lies in the relationship between the areas of these two squares: the smaller square's area is precisely one-quarter of the original square's area. Our mission is to embark on a mathematical journey to uncover the side length of Stacey's original square. This problem seamlessly blends geometric concepts with algebraic problem-solving, requiring us to translate the given information into mathematical equations and then employ algebraic techniques to arrive at the solution. We will navigate through the intricacies of area calculations, the elegance of quadratic equations, and the power of algebraic manipulation to unveil the hidden side length. Prepare to sharpen your pencils, engage your minds, and immerse yourselves in the captivating world of squares, areas, and the joy of mathematical deduction. This problem isn't just about finding a number; it's about understanding the relationships between geometric shapes and how algebraic tools can help us decipher those relationships. So, let's begin our exploration and unravel the mystery of Stacey's cloth!
Setting Up the Problem: Defining Variables and Equations
To effectively tackle this geometric puzzle, our initial step involves translating the word problem into the language of mathematics. This means assigning variables to the unknown quantities and formulating equations that accurately represent the given information. Let's denote the side length of Stacey's original square as 'x' inches. This is the key unknown we aim to determine. Since the cloth is a square, both its length and width are equal to 'x'. Consequently, the area of the original square can be expressed as x * x, or x². Now, let's consider the modifications Stacey makes. She cuts 3 inches off both the length and the width. This means the dimensions of the smaller square are (x - 3) inches by (x - 3) inches. The area of this smaller square, therefore, is (x - 3) * (x - 3), which can also be written as (x - 3)². The problem states a crucial relationship: the area of the smaller square is one-quarter (1/4) the area of the original square. We can translate this directly into an equation: (x - 3)² = (1/4) * x². This equation is the heart of our problem. It encapsulates the geometric relationship described in the problem statement in a concise algebraic form. Our next task is to manipulate this equation, employing algebraic techniques to isolate 'x' and find its value. This involves expanding the squared terms, simplifying the equation, and ultimately solving for 'x'. By carefully setting up the problem with clear variable definitions and a precise equation, we've laid a solid foundation for the subsequent steps in our solution.
Solving the Equation: A Step-by-Step Approach
With our equation, (x - 3)² = (1/4) * x², firmly established, we now embark on the crucial step of solving for 'x', which represents the side length of the original square. This involves a methodical application of algebraic principles. Our first move is to expand the left side of the equation, (x - 3)². This expansion yields x² - 6x + 9. So, our equation now stands as: x² - 6x + 9 = (1/4) * x². To simplify matters, let's eliminate the fraction. We can achieve this by multiplying both sides of the equation by 4. This gives us: 4(x² - 6x + 9) = x². Distributing the 4 on the left side, we get: 4x² - 24x + 36 = x². Now, our goal is to consolidate all the terms on one side of the equation, setting it equal to zero. This is a standard practice when dealing with quadratic equations. Subtracting x² from both sides, we obtain: 3x² - 24x + 36 = 0. We now have a quadratic equation in the standard form, ax² + bx + c = 0. Before we jump into solving it, let's see if we can simplify it further. Notice that all the coefficients (3, -24, and 36) are divisible by 3. Dividing the entire equation by 3, we arrive at a simpler form: x² - 8x + 12 = 0. This simplified quadratic equation is much easier to handle. Now, we have a few options for solving it: factoring, using the quadratic formula, or completing the square. In this case, factoring seems like the most straightforward approach. We need to find two numbers that multiply to 12 and add up to -8. A little thought reveals that -6 and -2 fit the bill perfectly. Therefore, we can factor the quadratic as: (x - 6)(x - 2) = 0. For this product to be zero, at least one of the factors must be zero. This gives us two possible solutions: x - 6 = 0 or x - 2 = 0. Solving these simple equations, we find x = 6 or x = 2. These are the potential side lengths of Stacey's original square.
Analyzing the Solutions: Finding the Valid Answer
Our algebraic journey has led us to two potential solutions for the side length of Stacey's original square: x = 6 inches and x = 2 inches. However, it's crucial to remember that mathematics often presents solutions that, while mathematically correct, might not make sense in the context of the real-world problem. This is precisely the situation we face now. We need to carefully analyze each solution and determine which one is valid in the context of the problem. Recall that Stacey cuts 3 inches off both the length and the width of the original square. This is a critical piece of information. If the original side length were 2 inches (x = 2), then cutting 3 inches off would result in a side length of 2 - 3 = -1 inches. A negative side length is geometrically impossible. Squares cannot have negative dimensions. Therefore, the solution x = 2 inches is not a valid solution in the context of our problem. It's a mathematically correct solution to the equation, but it doesn't represent a realistic scenario for the cloth. Now, let's consider the other solution: x = 6 inches. If the original side length were 6 inches, cutting 3 inches off would result in a side length of 6 - 3 = 3 inches. This is a positive, physically plausible side length. It aligns with the geometric constraints of the problem. Furthermore, if the original side length is 6 inches, the original area would be 6² = 36 square inches. The smaller square would have a side length of 3 inches and an area of 3² = 9 square inches. Indeed, 9 square inches is one-quarter of 36 square inches, satisfying the condition given in the problem. Therefore, x = 6 inches is a valid and meaningful solution. It represents the side length of Stacey's original square. In conclusion, while the algebraic solution yielded two possibilities, only one of them aligns with the geometric reality of the problem. This highlights the importance of not just solving equations but also interpreting the solutions within the context of the original problem.
Conclusion: The Side Length Revealed
Through a blend of geometric reasoning and algebraic problem-solving, we have successfully navigated the intricacies of Stacey's cloth puzzle. We began by translating the word problem into a mathematical equation, carefully defining variables and expressing the relationships between the areas of the squares. We then embarked on an algebraic journey, expanding, simplifying, and factoring our equation to arrive at two potential solutions. The critical step, however, was not just finding the solutions but also analyzing them within the context of the problem. This led us to discard one solution that, while mathematically sound, was geometrically impossible. Our analysis revealed that the only valid solution is x = 6 inches. Therefore, the side length of Stacey's original square was 6 inches. This problem serves as a beautiful illustration of how mathematics can be used to model and solve real-world scenarios. It showcases the power of algebra in representing geometric relationships and the importance of critical thinking in interpreting mathematical results. The process of solving this problem has reinforced our understanding of area calculations, quadratic equations, and the significance of considering the constraints of a problem when evaluating solutions. The journey through this puzzle has not only provided us with an answer but also with a deeper appreciation for the elegance and applicability of mathematics. The satisfaction of unraveling the mystery of Stacey's cloth lies not just in the final answer but in the logical and methodical process we undertook to arrive at it. So, the next time you encounter a geometric puzzle, remember the tools and techniques we've employed here, and embrace the challenge with confidence and curiosity.
