Solving For Width Using The Perimeter Formula A Step-by-Step Guide
Introduction
In this comprehensive guide, we will delve into the process of substituting given information into a formula and solving for the indicated variables. This is a fundamental skill in mathematics, particularly in algebra and geometry. We'll use the formula for the perimeter of a rectangle, which is $P=2L+2W$, where P represents the perimeter, L represents the length, and W represents the width. Our specific problem involves finding the width (W) when the perimeter (P) is 230 and the length (L) is expressed in terms of the width as L = W + 18. This type of problem demonstrates the power of algebraic manipulation to solve real-world problems. By understanding the principles involved, you'll be able to tackle similar problems with confidence. This guide provides a detailed walkthrough, ensuring clarity and understanding at every step. It's more than just solving an equation; it's about grasping the underlying mathematical concepts.
The formula P = 2L + 2W is a cornerstone in geometry, representing the total distance around a rectangle. Understanding this formula is crucial not only for solving mathematical problems but also for practical applications in everyday life, such as calculating the amount of fencing needed for a garden or the amount of trim required for a room. The variables in the equation each play a significant role. The perimeter, P, is the sum of all the sides of the rectangle. The length, L, is one of the longer sides, and the width, W, is the shorter side. In our problem, we're given that the length is related to the width by the equation L = W + 18. This relationship adds another layer of complexity, requiring us to substitute and simplify the equation to isolate the variable we're trying to solve for, which in this case, is the width. The process of substitution is a fundamental algebraic technique where one expression is replaced by another equivalent expression. This technique is not limited to geometry; it's used extensively in various areas of mathematics, including calculus and linear algebra. The ability to substitute and solve equations is a critical skill for anyone pursuing further studies in STEM fields or for anyone who wants to apply mathematical thinking to real-world problems.
Moreover, the problem introduces the concept of rounding answers to two decimal places, which is important for practical applications where precision is required but not absolute accuracy. Rounding ensures that the answer is presented in a manageable and meaningful way. Understanding when and how to round is a vital skill in mathematics and science. It's also essential to remember that rounding can introduce a small amount of error, so it's important to consider the level of precision required for the specific situation. In our problem, rounding to two decimal places will give us a practical answer for the width of the rectangle. We will use algebraic techniques such as substitution, simplification, and solving linear equations, to find the value of W. Each step will be explained in detail so you can follow the logic and process. This detailed guide will help you understand how to apply the formula, perform the substitution, and accurately solve for the width, making you more confident in your mathematical abilities.
Step-by-Step Solution
1. Understand the Given Information
We are given the formula for the perimeter of a rectangle: $P = 2L + 2W$. We also know that the perimeter, P, is 230, and the length, L, is related to the width, W, by the equation $L = W + 18$. Our goal is to find the value of W, which represents the width of the rectangle. The first step is to clearly understand the problem and identify what information we have and what we need to find. This involves carefully reading the problem statement and extracting the relevant details. In this case, we have the perimeter formula, the value of the perimeter, and a relationship between the length and the width. The key to solving this problem is to use the given information to create an equation that we can solve for the unknown variable, W. This involves substituting the given expressions into the formula and then using algebraic manipulation to isolate W. It's important to approach the problem systematically, breaking it down into smaller steps that are easier to manage. This not only helps in solving the problem correctly but also in understanding the underlying concepts. Remember, mathematics is not just about finding the answer; it's about understanding the process and the reasoning behind it. By taking the time to understand the given information, we set ourselves up for success in solving the problem. This initial step is crucial for building a solid foundation for the subsequent steps. In the following sections, we will see how to use this information to find the width of the rectangle.
Understanding the given information in a mathematical problem is like having a map before embarking on a journey. Without a clear understanding of where you are and where you need to go, the journey can become confusing and potentially unsuccessful. Similarly, in mathematics, a solid grasp of the givens—the known facts and relationships—is crucial for navigating the problem-solving process. In our case, we have the perimeter formula, which acts as our fundamental equation, the total perimeter value, which provides a quantitative anchor, and the length-width relationship, which introduces a crucial dependency between two key variables. This relationship is particularly significant because it allows us to express the length in terms of the width, paving the way for a single-variable equation. Recognizing this interdependence is a key step towards simplification. This initial phase of comprehension also involves identifying the ultimate goal: determining the width. Keeping this objective in mind helps to focus our efforts and direct our algebraic manipulations towards the desired outcome. Moreover, a thorough understanding of the given information can often reveal potential strategies for solving the problem. For instance, in this case, the presence of a formula and a variable relationship strongly suggests the method of substitution. It's also beneficial to consider the units involved, although in this abstract problem, units are not explicitly mentioned. However, in real-world applications, paying attention to units is essential for ensuring the validity and meaningfulness of the solution. In essence, a comprehensive understanding of the given information serves as the bedrock upon which the entire problem-solving process is built. It's the foundation that supports all subsequent steps, from formulating a strategy to executing calculations and interpreting results. Without this foundation, the entire endeavor risks becoming unstable and prone to errors. So, taking the time to thoroughly understand the givens is an investment that pays off significantly in the long run.
2. Substitute the Value of L into the Formula
Since we know that $L = W + 18$, we can substitute this expression for L in the perimeter formula: $P = 2L + 2W$. This substitution gives us $P = 2(W + 18) + 2W$. This step is crucial because it reduces the equation from having two unknowns (L and W) to having only one unknown (W). The process of substitution is a fundamental algebraic technique that allows us to solve equations with multiple variables. By replacing one variable with an equivalent expression, we simplify the equation and make it easier to solve. In this case, substituting W + 18 for L allows us to express the entire equation in terms of W. This is a significant step forward because we now have an equation that we can solve for W. The act of substitution is not just a mechanical process; it's a strategic move that simplifies the problem and brings us closer to the solution. It's like finding the right tool for the job – in this case, the right tool to simplify the equation. Before moving on, it's always a good idea to double-check the substitution to ensure that it has been done correctly. A small error in substitution can lead to a completely wrong answer. In our case, we have carefully replaced L with W + 18, so we can be confident that the substitution is correct. The next step will involve simplifying the equation by distributing and combining like terms. This will further isolate W and bring us closer to the final solution. The substitution is a bridge that connects the given information to the unknown we are trying to find. It's a powerful technique that is used extensively in mathematics and other fields. By mastering the art of substitution, you will be able to solve a wide range of problems with confidence.
Substitution in mathematics is akin to replacing a missing piece in a puzzle. When we substitute, we're not just plugging in numbers; we're replacing an unknown quantity or variable with an equivalent expression that we do know, thereby transforming the equation into a more manageable form. This is a powerful technique because it allows us to unravel complex relationships and find solutions that might otherwise remain hidden. In this specific context, substituting the expression for L (W + 18) into the perimeter formula is a strategic move that consolidates the equation into a single variable, W. This simplification is crucial because equations with a single variable are generally much easier to solve. The act of substitution itself requires careful attention to detail. It's essential to ensure that the replacement is made accurately, without any errors in copying or transcription. A small mistake during this stage can propagate through the rest of the solution, leading to an incorrect final answer. So, precision is paramount. Beyond the mechanical aspect of substitution, there's also a conceptual element to consider. Substitution highlights the interconnectedness of variables within an equation. It demonstrates how changing the value of one variable affects the others, and how these relationships can be leveraged to find solutions. This understanding is fundamental to mathematical reasoning and problem-solving. Furthermore, substitution is not just a technique confined to algebra; it's a versatile tool that appears across various branches of mathematics, from calculus to differential equations. Mastering this technique is therefore an investment in one's overall mathematical proficiency. In essence, substitution is a bridge that connects known information to unknown quantities. It's a fundamental step in the process of untangling complex equations and revealing the hidden values within. By mastering the art of substitution, we equip ourselves with a powerful tool for solving a wide array of mathematical problems.
3. Substitute the Value of P and Simplify the Equation
We know that $P = 230$, so we substitute this value into the equation: $230 = 2(W + 18) + 2W$. Now, we simplify the equation by distributing the 2 across the parentheses: $230 = 2W + 36 + 2W$. Next, we combine like terms: $230 = 4W + 36$. This step involves taking the equation we derived after substitution and simplifying it to make it easier to solve for W. Substituting the value of P allows us to move from a general equation to a specific equation that applies to our problem. The simplification process involves two key algebraic techniques: distribution and combining like terms. Distribution is the process of multiplying a term across a sum or difference inside parentheses. In our case, we distribute the 2 across the (W + 18), which gives us 2W + 36. This step eliminates the parentheses and makes it easier to combine like terms. Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. In our equation, we have two terms with W: 2W and 2W. Adding these together gives us 4W. By performing these simplification steps, we have transformed the equation from a more complex form to a simpler form: 230 = 4W + 36. This simplified equation is much easier to solve for W. It's like clearing away the clutter to reveal the essential parts of the equation. Each step in the simplification process is based on fundamental algebraic principles. Distribution is based on the distributive property of multiplication over addition, and combining like terms is based on the commutative and associative properties of addition. Understanding these principles is crucial for mastering algebraic manipulation. By carefully substituting the value of P and simplifying the equation, we have made significant progress towards solving for W. The next step will involve isolating W on one side of the equation, which will give us the final solution.
Substituting known values into an equation is a pivotal step in the problem-solving process, akin to plugging in the coordinates on a map to pinpoint our exact location. In our case, substituting the value of P (230) transforms the equation from a general relationship to a specific scenario, grounding the problem in concrete terms. This substitution allows us to move beyond the abstract and work with tangible numbers. The subsequent simplification of the equation is like clearing a path through a dense forest, making the destination—the solution—more visible and accessible. This involves applying algebraic techniques such as distribution and combining like terms, each of which serves to streamline the equation. Distribution, in essence, is the process of expanding a product, like unfolding a map to see the details. By distributing the 2 across the parentheses, we eliminate the grouping and reveal the individual components of the expression. Combining like terms, on the other hand, is like organizing our supplies and equipment before a journey, ensuring that everything is in its proper place and readily available. By grouping together terms with the same variable, we consolidate the equation and reduce its complexity. The goal of these simplification techniques is to isolate the variable we're trying to solve for, in this case, W. By stripping away the extraneous elements and streamlining the equation, we make it easier to manipulate and ultimately solve for the unknown. Each step in the simplification process should be performed with care and precision, ensuring that no errors are introduced along the way. A mistake at this stage can derail the entire solution. In essence, substituting the known value and simplifying the equation is like setting the stage for the final act of the problem-solving process. It's the crucial preparatory work that lays the foundation for finding the solution with confidence and accuracy.
4. Isolate the Variable W
To isolate W, we first subtract 36 from both sides of the equation: $230 - 36 = 4W + 36 - 36$, which simplifies to $194 = 4W$. Then, we divide both sides by 4: $\frac{194}{4} = \frac{4W}{4}$, which gives us $W = 48.5$. This stage is where we bring the variable we're solving for, in this case, W, into solitary confinement. Isolating a variable is a fundamental technique in algebra, and it's the key to solving for its value. The principle behind isolating a variable is based on the properties of equality. We can perform the same operation on both sides of an equation without changing its balance. In this case, we first subtract 36 from both sides. This eliminates the constant term on the side with W, bringing us closer to isolating it. Subtracting the same value from both sides maintains the equality, ensuring that the equation remains valid. The next step is to divide both sides by the coefficient of W, which is 4. This undoes the multiplication and leaves W by itself on one side of the equation. Again, dividing both sides by the same value maintains the equality. The result of these two operations is that we have successfully isolated W, and we have found its value: W = 48.5. This is the solution to the problem. It's important to remember that each step in the isolation process is based on sound algebraic principles. We are not just randomly manipulating the equation; we are applying specific operations that preserve the equality and bring us closer to the solution. Isolating a variable is like peeling away the layers of an onion to reveal the core. Each step removes a layer of complexity, until we are left with the variable itself. This technique is used extensively in mathematics and other fields. By mastering the art of isolating variables, you will be able to solve a wide range of equations and problems with confidence.
Isolating the variable in an equation is like freeing a prisoner from their cell. It's the crucial step that allows us to determine the value of the unknown and unlock the solution to the problem. This process hinges on the fundamental principle of maintaining balance within the equation. Whatever operation we perform on one side, we must mirror it on the other side, ensuring that the equality remains intact. In our case, the first step towards isolating W involves neutralizing the constant term (+36) that's sharing its space. We achieve this by subtracting 36 from both sides of the equation, effectively canceling out the constant on the right-hand side. This is akin to removing an obstacle that's blocking our path. The next step is to address the coefficient (4) that's multiplying W. To liberate W completely, we divide both sides of the equation by 4, which effectively undoes the multiplication and leaves W standing alone. This is like cutting the final chain that binds the prisoner. Each of these operations—subtraction and division—is a precise algebraic maneuver designed to strategically peel away the layers surrounding the variable. It's a delicate dance of mathematical manipulation, where each move must be carefully calculated to maintain the equation's integrity. The ultimate goal is to have W isolated on one side, with its value clearly revealed on the other side. This isolated value represents the solution to the problem, the answer we've been seeking. Isolating the variable is not just a mechanical process; it's a demonstration of our understanding of algebraic principles and our ability to apply them strategically. It's a testament to our problem-solving skills and our capacity to navigate the complexities of mathematical equations. By mastering this technique, we equip ourselves with a powerful tool for tackling a wide range of mathematical challenges.
5. Round the Answer to Two Decimal Places (if necessary)
In this case, the answer, $W = 48.5$, is already given to one decimal place. Since we need to round to two decimal places as necessary, we can write it as $W = 48.50$. This step emphasizes the importance of paying attention to the instructions and providing the answer in the specified format. Rounding is a common practice in mathematics and other fields, especially when dealing with decimal numbers. It allows us to express the answer with a certain level of precision, depending on the context of the problem. In this case, the instructions specifically ask us to round answers to two decimal places as necessary. This means that if the answer has more than two decimal places, we need to round it to two decimal places. However, if the answer has fewer than two decimal places, we can simply add zeros to the end to make it have two decimal places. In our case, the answer W = 48.5 has only one decimal place. To round it to two decimal places, we simply add a zero to the end, which gives us W = 48.50. This does not change the value of the answer, but it ensures that the answer is presented in the correct format. Rounding is not just a cosmetic step; it's an important part of communicating the answer clearly and accurately. It also reflects our understanding of significant figures and the level of precision required for the problem. By rounding the answer to two decimal places, we are indicating that we have calculated the answer to the nearest hundredth. This level of precision is often sufficient for practical applications. Rounding also helps to avoid unnecessary clutter in the answer. If we were to write the answer with many decimal places, it would be difficult to read and understand. Rounding makes the answer more concise and easier to interpret. In conclusion, rounding to the specified number of decimal places is an essential step in the problem-solving process. It ensures that the answer is presented in the correct format and with the appropriate level of precision.
Rounding is like fine-tuning an instrument to achieve the desired pitch. It's the process of adjusting a numerical value to a specified level of precision, ensuring that it aligns with the context of the problem and the required level of detail. In our case, the instruction to round to two decimal places serves as a guideline for presenting the final answer in a clear and meaningful way. Rounding is not just about chopping off digits; it's about representing a value with the appropriate level of accuracy. When we round, we're making a conscious decision about which digits are significant and which can be discarded without compromising the integrity of the result. In this particular instance, our initial calculation yielded a value for W with one decimal place (48.5). To adhere to the instruction of rounding to two decimal places, we simply add a zero to the end, transforming it to 48.50. This addition doesn't alter the value itself; it merely expresses it with the required level of precision. Rounding is particularly crucial when dealing with real-world measurements, where absolute precision is often unattainable or unnecessary. In such situations, rounding allows us to present the data in a manageable and interpretable format. It also helps to avoid the illusion of greater accuracy than is warranted by the measurement process. Furthermore, rounding plays a vital role in simplifying calculations and reducing the risk of error propagation. By working with rounded values, we can often streamline the computational process without significantly affecting the final outcome. However, it's essential to be mindful of the potential for rounding errors to accumulate, especially in multi-step calculations. In summary, rounding is a fundamental skill in mathematics and science, allowing us to express numerical values with the appropriate level of precision and clarity. It's a tool that helps us to communicate results effectively and avoid unnecessary complexity.
Final Answer
Therefore, the width, $W = 48.50$. This concludes our step-by-step solution to finding the width of the rectangle given the perimeter and the relationship between the length and the width. The final answer, W = 48.50, is the result of a series of logical and algebraic steps. We started by understanding the given information, which included the perimeter formula, the value of the perimeter, and the relationship between the length and the width. We then used the technique of substitution to eliminate one of the variables and create an equation with only one unknown. This allowed us to simplify the equation and isolate the variable we were trying to solve for. Finally, we rounded the answer to the specified number of decimal places to ensure that it was presented in the correct format. The process of solving this problem illustrates the power of algebraic manipulation and the importance of following a systematic approach. Each step in the solution is based on sound mathematical principles, and by carefully applying these principles, we were able to arrive at the correct answer. This problem also highlights the connection between algebra and geometry. The perimeter formula is a geometric concept, but solving for the width requires algebraic techniques. This connection between different branches of mathematics is a common theme, and understanding these connections is crucial for developing a deeper understanding of mathematics. The final answer, W = 48.50, is not just a number; it's the solution to a real-world problem. It represents the width of a rectangle with a specific perimeter and a specific relationship between its length and width. This problem demonstrates how mathematics can be used to model and solve problems in the real world.
The final answer, W = 48.50, is more than just a numerical result; it's the culmination of a systematic problem-solving journey. It represents the width of the rectangle, a concrete dimension derived from the abstract relationships defined by the perimeter formula and the length-width equation. This solution is a testament to the power of algebraic manipulation and the importance of a structured approach to problem-solving. Our journey began with a clear understanding of the given information: the perimeter formula (P = 2L + 2W), the numerical value of the perimeter (P = 230), and the relationship between length and width (L = W + 18). Each piece of information served as a vital clue, guiding us towards the solution. The strategic use of substitution allowed us to consolidate the equations, transforming a two-variable problem into a single-variable equation. This simplification was a crucial step, paving the way for us to isolate W and determine its value. The subsequent algebraic manipulations, including distribution, combining like terms, and isolating the variable, were executed with precision and care, ensuring that the integrity of the equation was maintained throughout the process. Finally, the act of rounding the answer to two decimal places (as necessary) demonstrated our attention to detail and adherence to the problem's instructions. This final touch ensured that the solution was presented in the appropriate format and with the required level of precision. The entire process, from initial comprehension to final answer, underscores the interconnectedness of mathematical concepts. Algebra and geometry, often perceived as distinct branches, seamlessly intertwined in this problem, highlighting the unifying nature of mathematics. The final answer, therefore, represents not only the width of the rectangle but also a successful application of mathematical principles and problem-solving strategies. It's a tangible outcome that reinforces the value of mathematical thinking and its relevance to real-world scenarios.
In conclusion, solving for width in a perimeter formula involves a series of algebraic steps, including substitution, simplification, and isolation of the variable. By following these steps carefully, we can accurately determine the width of the rectangle. Remember to always double-check your work and round your answers as necessary. With practice, you'll become more confident in your ability to solve similar problems. This structured approach is not just applicable to mathematical problems but can be extended to various problem-solving scenarios in daily life.
