Solving For Square Window Dimensions A Step-by-Step Guide
Understanding the relationship between the sides and areas of squares is crucial in solving this geometric problem. This article breaks down the problem of finding the sides of two square windows given their combined area and the difference in their side lengths. We'll walk through the solution step-by-step, explaining the underlying algebraic principles and offering insights into how to approach similar problems. Our goal is not only to provide the answer but also to ensure a clear understanding of the problem-solving process. Let’s dive into this fascinating mathematical exploration!
Problem Statement
The problem presents a classic scenario involving geometric shapes and algebraic relationships. We are given that the combined area of two square windows is 1,025 square inches. A key piece of information is that the larger window's sides are 5 inches longer than the sides of the smaller window. The challenge is to determine the side length of the smaller window. This requires translating the word problem into mathematical equations and solving for the unknown. The beauty of this problem lies in its simplicity and the elegant application of basic geometric and algebraic principles. It showcases how real-world scenarios can be modeled mathematically, allowing us to find precise solutions through logical reasoning and calculations.
Setting up the Equations
To effectively solve this problem, we need to translate the given information into mathematical equations. Let's denote the side length of the smaller window as 's' inches. Since the larger window's sides are 5 inches longer, its side length will be 's + 5' inches. Remembering the formula for the area of a square (Area = side * side or A = s²), the area of the smaller window is s², and the area of the larger window is (s + 5)². The problem states that the combined area of the two windows is 1,025 square inches. This leads us to the fundamental equation: s² + (s + 5)² = 1025. This equation is the cornerstone of our solution, encapsulating the geometric relationships and the given total area. It is a quadratic equation, which means it involves a squared term (s²), and solving it will require us to use algebraic techniques such as expanding, simplifying, and potentially using the quadratic formula or factoring. Mastering the art of translating word problems into algebraic equations is a critical skill in mathematics, as it allows us to apply the power of algebra to solve a wide range of practical and theoretical problems.
Solving the Quadratic Equation
Now that we have our equation, s² + (s + 5)² = 1025, the next step is to solve it for 's'. This involves a series of algebraic manipulations to isolate 's' and find its value. First, we expand the term (s + 5)² which gives us s² + 10s + 25. Substituting this back into our equation, we get s² + s² + 10s + 25 = 1025. Combining like terms, we simplify the equation to 2s² + 10s + 25 = 1025. To further simplify, we subtract 1025 from both sides, resulting in the quadratic equation 2s² + 10s - 1000 = 0. Before attempting to factor or use the quadratic formula, we can simplify the equation by dividing all terms by 2, which gives us s² + 5s - 500 = 0. This simplified quadratic equation is now in a standard form that is easier to work with. We can now proceed to solve it by factoring, using the quadratic formula, or completing the square. Each of these methods has its own advantages, and the choice of method often depends on the specific equation and personal preference. In this case, factoring might be a viable option if we can find two numbers that multiply to -500 and add up to 5. Alternatively, the quadratic formula provides a direct solution, and completing the square can be useful for understanding the structure of the equation. The process of solving quadratic equations is a fundamental skill in algebra, with applications in various fields ranging from physics and engineering to economics and finance.
Factoring the Quadratic
The simplified quadratic equation we need to solve is s² + 5s - 500 = 0. Factoring is a powerful technique for solving quadratic equations, especially when the coefficients are integers and the roots are rational numbers. The key to factoring a quadratic expression of the form ax² + bx + c is to find two numbers that multiply to 'ac' and add up to 'b'. In our case, we need two numbers that multiply to -500 (1 * -500) and add up to 5. After some thought, we can identify those numbers as 25 and -20. Indeed, 25 * -20 = -500 and 25 + (-20) = 5. Now we can rewrite the middle term (5s) using these two numbers: s² + 25s - 20s - 500 = 0. Next, we factor by grouping. We group the first two terms and the last two terms: (s² + 25s) + (-20s - 500) = 0. Now we factor out the greatest common factor from each group. From the first group, we factor out 's', giving us s(s + 25). From the second group, we factor out '-20', giving us -20(s + 25). Notice that we now have a common factor of (s + 25). Factoring this out, we get (s + 25)(s - 20) = 0. Now we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible equations: s + 25 = 0 or s - 20 = 0. Solving these equations, we find two possible values for 's': s = -25 or s = 20. However, since 's' represents the side length of a window, it cannot be negative. Therefore, we discard the solution s = -25. This leaves us with s = 20 as the only valid solution. Factoring is a valuable skill in algebra, allowing us to simplify expressions, solve equations, and gain deeper insights into mathematical relationships.
Determining the Valid Solution
From the factoring process, we arrived at two possible solutions for 's': s = -25 and s = 20. However, in the context of our problem, 's' represents the side length of a window. A fundamental principle of geometry is that lengths cannot be negative. Therefore, the solution s = -25 is not physically meaningful and must be discarded. This leaves us with s = 20 as the only valid solution for the side length of the smaller window. This step highlights the importance of considering the context of a problem when interpreting mathematical solutions. Mathematical equations and formulas can sometimes produce results that, while mathematically correct, do not make sense in the real world. Being able to identify and reject such solutions is a crucial skill in applying mathematics to practical problems. In this case, the negative solution arose from the algebraic manipulation, but it was the geometric constraint (side lengths must be positive) that allowed us to choose the correct answer. This interplay between mathematical calculations and real-world constraints is a hallmark of applied mathematics and problem-solving.
Final Answer
Therefore, the side length of the smaller window is 20 inches. This solution is consistent with the problem statement and makes logical sense in the context of the physical situation. We arrived at this answer by carefully translating the word problem into algebraic equations, solving the resulting quadratic equation through factoring, and considering the physical constraints of the problem. The side length of the smaller window is 20 inches, which corresponds to option B in the multiple-choice answers provided. To verify our solution, we can calculate the side length of the larger window, which is s + 5 = 20 + 5 = 25 inches. Then, we can calculate the areas of the two windows: the smaller window has an area of 20² = 400 square inches, and the larger window has an area of 25² = 625 square inches. The combined area is 400 + 625 = 1025 square inches, which matches the given information in the problem. This verification step confirms the accuracy of our solution and demonstrates a thorough approach to problem-solving. The process of solving this problem highlights the importance of several key mathematical skills, including translating word problems into equations, manipulating algebraic expressions, solving quadratic equations, and interpreting solutions in context.
Conclusion
In summary, this problem elegantly demonstrates the application of mathematical principles to solve real-world scenarios. By carefully setting up equations, applying algebraic techniques, and considering geometric constraints, we successfully determined the side length of the smaller window to be 20 inches. This step-by-step solution not only provides the answer but also illuminates the underlying problem-solving process. Problems like these underscore the importance of a solid foundation in algebra and geometry, as well as the ability to translate abstract mathematical concepts into practical solutions. The process of translating word problems into equations, solving those equations, and interpreting the solutions within the context of the problem is a fundamental skill in mathematics and its applications. It is a skill that is honed through practice and a deep understanding of mathematical principles. As we have shown in this detailed solution, with a clear understanding of the problem, the right tools, and a systematic approach, even seemingly complex problems can be solved effectively and efficiently. This problem serves as a valuable exercise in mathematical reasoning and problem-solving, showcasing the power and versatility of mathematics in addressing real-world challenges. The skills learned in solving this problem can be applied to a wide range of similar problems in mathematics, science, and engineering.
