Unveiling Sabrina's Rug Border: A Mathematical Exploration
Hey there, math enthusiasts! Let's dive into a fun and engaging problem about Sabrina and her rug. This isn't just about finding an answer; it's about understanding the journey, the math behind it all. We'll break down the problem step-by-step, making sure it's super clear and easy to follow. So, grab your notebooks and let's get started on figuring out Sabrina's rug border!
Understanding the Problem: The Rug's Transformation
Okay, so the scene is set: Sabrina has a rug, and this rug has a perimeter of 18 feet. Now, a perimeter, as you probably know, is just the total distance around the outside of a shape. Think of it like walking around the rug; the perimeter is how far you walk. The fun begins when Sabrina decides to jazz up her rug with a border. This border isn't just for looks; it changes the rug's dimensions. After adding the border, the rug's new length becomes 8 feet, and the new width is 4 feet. The crucial detail is that the border has the same width all the way around the rug. This consistency is super important when we start solving the problem. The question we need to answer is: what's the equation that represents x, the width of the border? We're not just looking for the answer; we want to see the equation that makes it all click. The keywords here are perimeter, border, length, and width. The problem involves a bit of algebra, a little geometry, and a dash of real-world application. It's the kind of problem that makes math exciting because it shows how math applies to everyday things, like decorating your home. We're going to break down the rug's original dimensions and how the border alters them to help us establish a concrete understanding. This step will enable us to formulate the right equation to figure out the border's width, which will be the x we're after. The key is to remember that the border expands the rug on all sides equally.
Breaking Down the Original Rug
Before Sabrina added the border, her rug had a perimeter of 18 feet. We know that the perimeter of a rectangle is calculated using the formula: P = 2l + 2w, where l is the length, and w is the width. So, we have 18 = 2l + 2w. This formula is your foundation; understanding it is key. We need to figure out what the original length and width were before the border. However, we have a problem: we only have the perimeter, so we can't pinpoint the exact length and width. That's fine; we'll come back to this. Keep in mind that when we add the border, both the length and the width will increase. Because the border's width is the same on all sides, the original length is going to increase by 2x (because the border adds width on both sides of the length), and the original width is going to increase by 2x as well. Since we know the new dimensions are 8 feet by 4 feet, we can work backward from there. The beauty of this problem is that it brings together the original rug, the effect of the border, and the new dimensions into one cohesive package. This highlights the importance of setting up equations to explain a problem step by step. When we solve the equation, we'll find the value of x - the width of the border. It's like a puzzle where each step has a reason and a purpose.
The Border's Impact on Dimensions
So, Sabrina adds a border of equal width all around the rug. This addition affects the length and width of the rug. The new length is 8 feet, and the new width is 4 feet. Now, let's consider this logically: before the border, the rug was shorter in both dimensions. The border added to both sides of the rug. Because the border goes around the entire rug, it adds to both the length and the width. If we call the width of the border x, the original length must have been 8 - 2x and the original width 4 - 2x (subtracting x from each end of the length and width). This is because the new length and width include the border's width. For the length, x is added to both ends, so the original length is the new length minus 2 times the width of the border. The same applies to the width. The goal here is to come up with an equation that represents the original perimeter, which we know is 18 feet. Using the formula P = 2l + 2w, and the original perimeter being 18 feet, we can substitute the expressions for the original length and width. This allows us to create an equation with only x, the width of the border, as the unknown. The equation we will form will allow us to mathematically find the original length and width of the rug, providing all the needed context for our final step.
Formulating the Equation: Putting it all Together
Alright, it's time to build the equation that represents x, the width of the border. We know the original perimeter was 18 feet, and we also know the relationship between the original dimensions and the new dimensions with the border. The new length is 8 feet, and the new width is 4 feet. Let's use x to represent the width of the border. As we discussed, the original length can be represented as 8 - 2x (subtracting x from each end). Similarly, the original width is 4 - 2x. We go back to the original formula: P = 2l + 2w. Now, substitute the original length and width, using the new dimensions and the border width x: 18 = 2(8 - 2x) + 2(4 - 2x). Let's take a moment to look at this equation. On the left side, we have the original perimeter: 18. On the right side, we have the original dimensions expressed in terms of x. The equation shows that the original perimeter (18 feet) is equal to twice the original length plus twice the original width. The equation elegantly captures all the information we have, relating the original perimeter, the new dimensions, and the border's width. Simplifying this equation is our next task. Doing so will allow us to isolate x and find the actual width of the border. The equation is the centerpiece of the problem-solving strategy, and a clear understanding of its components is essential for the next steps.
Simplifying and Solving the Equation
We have the equation: 18 = 2(8 - 2x) + 2(4 - 2x). Let's simplify and solve for x. First, distribute the 2 across the terms within the parentheses: 18 = 16 - 4x + 8 - 4x. Now, combine the constants: 18 = 24 - 8x. Next, isolate the term with x. Subtract 24 from both sides: 18 - 24 = -8x. This simplifies to -6 = -8x. Now, to solve for x, divide both sides by -8: x = -6 / -8. The negative signs cancel out, leaving us with x = 3/4 or 0.75. So, the width of the border, x, is 0.75 feet or 9 inches. This is how we can determine the exact width of the border. The process included simplifying, isolating x, and solving for its value. The method showcases the power of algebra to solve for unknowns by arranging terms systematically. That final value gives us the precise width of the rug border, bringing us to a complete and satisfying solution.
The Answer and What It Means
So, to recap, the equation that represents x, the width of the border, is 18 = 2(8 - 2x) + 2(4 - 2x), and the width of the border x is 0.75 feet or 9 inches. This means that Sabrina added a border that is 9 inches wide all the way around her rug. This mathematical problem wasn't just about finding the answer. It's about seeing how the perimeter, length, and width all interact. Understanding this is super important. We started with the original rug's perimeter, saw how the border changed the dimensions, and used algebra to find the border's width. The equation is more than just a formula; it's a representation of how the border affected the rug's size. By solving the equation, we found the unknown variable x which is the key to solving the problem. This problem showed us how math is used in real-life, like decorating. Now, imagine if Sabrina needed to buy more border material. She'd use this information to make sure she had enough. We took a common situation and turned it into a math lesson. It's a prime example of the beauty of math: it's practical, logical, and always helpful. So, keep up the great work, and keep exploring the amazing world of mathematics! The key is to take the problems, break them down, and understand the logic step by step.
Key Takeaways and Insights
Here are the important things to remember from this problem:
- Perimeter and Dimensions: The perimeter is the total distance around a shape, while length and width define the shape's size. Both are important.
- Impact of a Border: Adding a border affects the length and width of the original shape.
- Formulating Equations: Using the correct formulas, like P = 2l + 2w, is crucial for setting up the right equations.
- Solving for Unknowns: Algebra helps us find unknown variables like x, which in our case was the border's width.
- Real-World Application: Math is everywhere! This problem showcases how we can use math in everyday situations.
Keep these insights in mind, and you'll be well-prepared to tackle any math problem that comes your way! Remember, the more you practice, the easier and more fun math becomes. And most importantly, always try to understand why the math works, not just how it works. Happy calculating!
