Finding The Length Of A Rectangle With A Perimeter Of 42 Inches
In the realm of geometry, rectangles stand as fundamental shapes characterized by their two pairs of equal-length sides and four right angles. Calculating the perimeter of a rectangle is a common mathematical task, often encountered in everyday situations like fencing a garden or framing a picture. The perimeter, in essence, is the total distance around the rectangle, obtained by summing the lengths of all its sides. To delve deeper into the concept, the perimeter ( extit{P}) of a rectangle can be mathematically expressed using the formula extit{P} = 2 extit{l} + 2 extit{w}, where extit{l} represents the length and extit{w} signifies the width. This formula stems from the fact that a rectangle has two sides of equal length and two sides of equal width. When presented with a problem involving the perimeter and one of the dimensions, we can manipulate this formula to solve for the unknown dimension. This involves substituting the known values into the formula and then using algebraic techniques to isolate the variable representing the unknown dimension. For example, if we know the perimeter and the width, we can rearrange the formula to solve for the length, and vice versa. Mastering this fundamental concept is crucial for tackling a wide range of geometric problems and real-world applications. Understanding the relationship between perimeter, length, and width allows us to make informed decisions in various practical scenarios, such as determining the amount of material needed for construction projects or calculating the distance around a room. The ability to apply this formula effectively showcases a solid grasp of geometric principles and problem-solving skills.
The mathematical challenge at hand presents us with a classic geometry problem: determining the length of a rectangle given its perimeter and width. The problem states that the perimeter of the rectangle is 42 inches, and its width is 6 inches. Our objective is to find the length of this rectangle. To tackle this problem effectively, we can employ a step-by-step approach, leveraging the perimeter formula we discussed earlier. First, we need to recall the formula for the perimeter of a rectangle, which is extitP} = 2 extit{l} + 2 extit{w}, where extit{P} is the perimeter, extit{l} is the length, and extit{w} is the width. Next, we substitute the given values into the formula. We know that extit{P} = 42 inches and extit{w} = 6 inches. Substituting these values, we get the equation 42 = 2 extit{l} + 2(6). Now, we simplify the equation by performing the multiplication + 12. To isolate the term with the unknown variable ( extitl}), we subtract 12 from both sides of the equation + 12 - 12, which simplifies to 30 = 2 extitl}. Finally, to solve for extit{l}, we divide both sides of the equation by 2 / 2, which gives us extit{l} = 15 inches. Therefore, the length of the rectangle is 15 inches. This step-by-step approach demonstrates how we can systematically solve geometry problems by breaking them down into smaller, manageable steps. By carefully applying the relevant formulas and using algebraic techniques, we can arrive at the correct solution.
To pinpoint the length of the rectangle, we embark on a mathematical journey, utilizing the formula that connects the perimeter, length, and width of a rectangle. As we established earlier, the perimeter ( extit{P}) is calculated using the formula extit{P} = 2 extit{l} + 2 extit{w}, where extit{l} represents the length and extit{w} denotes the width. In this specific scenario, we are provided with the perimeter ( extit{P} = 42 inches) and the width ( extit{w} = 6 inches), and our mission is to determine the length ( extit{l}). The initial step involves substituting the known values into the formula, transforming it into an equation with a single unknown variable. This substitution yields the equation 42 = 2 extit{l} + 2(6). Next, we simplify the equation by performing the multiplication, resulting in 42 = 2 extit{l} + 12. Our objective is to isolate the term containing the unknown variable ( extit{l}) on one side of the equation. To achieve this, we subtract 12 from both sides of the equation, maintaining the balance and equality. This operation transforms the equation into 42 - 12 = 2 extit{l} + 12 - 12, which simplifies to 30 = 2 extit{l}. Finally, to solve for extit{l}, we divide both sides of the equation by 2, effectively isolating extit{l}. This division results in 30 / 2 = 2 extit{l} / 2, which simplifies to extit{l} = 15 inches. Therefore, the length of the rectangle is 15 inches. This meticulous mathematical process exemplifies how we can systematically solve for an unknown dimension in a geometric problem by applying the appropriate formula and employing algebraic manipulations.
After meticulously dissecting the problem and applying the principles of geometry, we arrive at the definitive answer: the length of the rectangle is 15 inches. This corresponds to option B in the provided choices. Let's delve into why this is the correct answer and why the other options are incorrect. We began with the formula for the perimeter of a rectangle, extit{P} = 2 extit{l} + 2 extit{w}, and substituted the given values of the perimeter (42 inches) and width (6 inches) into the equation. This resulted in the equation 42 = 2 extit{l} + 2(6). Through a series of algebraic manipulations, including simplification, subtraction, and division, we isolated the variable representing the length ( extit{l}) and found its value to be 15 inches. Option A, 12 inches, is incorrect because it would result in a perimeter less than 42 inches when used in the formula. Specifically, if the length were 12 inches, the perimeter would be 2(12) + 2(6) = 24 + 12 = 36 inches, which does not match the given perimeter of 42 inches. Option C, 21 inches, is also incorrect. If the length were 21 inches, the perimeter would be 2(21) + 2(6) = 42 + 12 = 54 inches, which exceeds the given perimeter of 42 inches. Option D, 40 inches, is significantly larger than the correct length and would result in an even greater perimeter. If the length were 40 inches, the perimeter would be 2(40) + 2(6) = 80 + 12 = 92 inches, which is far from the given perimeter of 42 inches. Therefore, option B, 15 inches, is the only option that satisfies the given conditions and accurately represents the length of the rectangle. This meticulous verification process underscores the importance of not only arriving at a solution but also confirming its validity and understanding why other options are incorrect.
The concept of perimeter extends far beyond the confines of the classroom and finds its application in a multitude of real-world scenarios. Understanding how to calculate the perimeter of various shapes, particularly rectangles, is a valuable skill that can be applied in numerous practical situations. One common application is in home improvement and construction projects. When building a fence around a yard or a garden, calculating the perimeter is crucial for determining the amount of fencing material needed. Similarly, when framing a picture or installing baseboards in a room, knowing the perimeter helps in accurately cutting and fitting the materials. In interior design, perimeter calculations play a vital role in space planning. Determining the perimeter of a room can help in deciding the placement of furniture and ensuring that there is enough space for movement. For instance, calculating the perimeter of a rug can help in selecting the right size for a room. In gardening and landscaping, perimeter calculations are essential for planning the layout of flower beds, vegetable patches, or walkways. Knowing the perimeter of a garden bed helps in determining the amount of edging material required and in estimating the number of plants that can be accommodated. In sports and recreation, perimeter calculations are used in designing and marking playing fields, such as football fields, basketball courts, and tennis courts. The dimensions of these fields, including their perimeters, are precisely defined to ensure fair play and accurate measurements. In manufacturing and packaging, perimeter calculations are used in determining the size of packaging materials, such as boxes and cartons. Knowing the perimeter of an object helps in designing a package that can securely hold the object and minimize waste. These examples illustrate the diverse and practical applications of perimeter calculations in various fields. Mastering this concept not only enhances our mathematical skills but also equips us with the ability to solve real-world problems efficiently and effectively. The ability to apply perimeter calculations in practical situations showcases the relevance and importance of mathematics in our daily lives.
Conclusion
In summary, we've successfully navigated through the problem of finding the length of a rectangle given its perimeter and width. By understanding the fundamental formula for the perimeter of a rectangle, extit{P} = 2 extit{l} + 2 extit{w}, and employing a step-by-step problem-solving approach, we were able to accurately determine the length to be 15 inches. This journey not only reinforces our grasp of geometric concepts but also highlights the practical applications of perimeter calculations in various real-world scenarios. From home improvement projects to sports field design, the ability to calculate perimeters proves to be a valuable skill. The meticulous process of substituting known values, simplifying equations, and isolating the unknown variable demonstrates the power of algebraic techniques in solving geometric problems. Furthermore, the verification of the solution and the elimination of incorrect options underscore the importance of critical thinking and attention to detail in mathematics. This problem serves as a reminder that mathematics is not merely a collection of formulas and equations but a powerful tool for understanding and interacting with the world around us. By mastering fundamental concepts and developing problem-solving skills, we can confidently tackle a wide range of mathematical challenges and apply our knowledge to real-life situations. The ability to approach problems systematically, break them down into smaller steps, and apply the appropriate formulas is a hallmark of mathematical proficiency. As we continue our exploration of mathematics, we will encounter more complex problems and concepts, but the foundation we have built by mastering fundamental principles will serve us well. The journey of mathematical discovery is ongoing, and each problem we solve contributes to our overall understanding and appreciation of this fascinating field.
