Rectangle Dimensions: Solve For Width!
Introduction
Hey guys! Let's dive into a fun math problem today that involves rectangles, perimeters, and a little bit of algebra. We've got a classic geometry challenge on our hands, and we're going to break it down step by step so everyone can follow along. The problem states that the length of a rectangle is three times its width plus 5 cm, and the perimeter of the rectangle is 42 cm. Our mission, should we choose to accept it (and we do!), is to figure out the width of this rectangle. So, grab your thinking caps, and let’s get started! To begin, it's important to understand the basic properties of a rectangle. A rectangle is a four-sided shape with opposite sides that are equal in length and four right angles. The perimeter of a rectangle is the total distance around its outside, which we can find by adding up the lengths of all four sides. In our problem, we're given the perimeter (42 cm), and we have a relationship between the length and the width. This is where the algebra comes in handy! We can use variables to represent the unknown dimensions and set up an equation to solve for the width. This is a super useful skill, not just for math class, but for all sorts of real-world situations where we need to figure out measurements and dimensions. For instance, if you're planning to build a fence around your backyard or put up wallpaper in a room, understanding how to calculate perimeters and areas is essential. So, let’s tackle this problem with confidence and see how we can unlock the mystery of this rectangle's dimensions.
Understanding the Problem
Okay, first things first, let's really understand what the problem is asking. We know the length of the rectangle is related to its width, and we know the perimeter. The key here is to translate the words into mathematical expressions. When we say the “length is three times the width plus 5 cm,” we're setting up a direct relationship between these two dimensions. We can use a variable, let's say 'w', to represent the width. Then, the length can be expressed as '3w + 5'. This is a crucial step because it allows us to work with algebraic equations instead of just words. Next, we know the perimeter of the rectangle is 42 cm. Remember, the perimeter is the sum of all the sides. A rectangle has two lengths and two widths, so we can write the formula for the perimeter as P = 2l + 2w. Now we're getting somewhere! We have a formula, we have an expression for the length in terms of the width, and we have the given perimeter. This is like having all the pieces of a puzzle laid out in front of us. Now, the fun part is putting them together. Why is this important? Well, breaking down a problem into smaller, manageable parts is a fantastic strategy for solving all sorts of challenges, not just in math. In everyday life, whether you're planning a trip, budgeting your finances, or organizing a project at work, the ability to identify the key information and understand the relationships between different factors is super helpful. So, by practicing these skills with math problems, you're actually building valuable problem-solving muscles that you can use in all areas of your life. Let's keep going and see how we can use these pieces to find the width of our rectangle.
Setting up the Equation
Alright, guys, now for the juicy part – setting up the equation! We've already laid the groundwork by understanding the problem and identifying the key relationships. We know that the perimeter (P) is 42 cm, and we have the formula for the perimeter of a rectangle: P = 2l + 2w. We also know that the length (l) can be expressed as 3w + 5. Now, we can substitute the expression for the length into the perimeter formula. This is where the magic happens! By substituting, we get: 42 = 2(3w + 5) + 2w. See what we did there? We replaced the 'l' in the perimeter formula with '3w + 5'. Now we have an equation with just one variable, 'w', which represents the width we're trying to find. This is a significant step because we've transformed a word problem into an algebraic equation that we can solve. Think of it like translating a sentence from one language to another. We've taken the information given in words and translated it into the language of math. This is a powerful skill in mathematics and other fields. The ability to represent real-world situations with equations allows us to use mathematical tools to find solutions. Why is this so cool? Because it gives us a structured way to solve problems. Instead of just guessing or trying different things randomly, we have a clear process: identify the relationships, write an equation, and solve for the unknown. This is the essence of algebra, and it's a cornerstone of many scientific and engineering disciplines. So, now that we have our equation, 42 = 2(3w + 5) + 2w, the next step is to solve it. Are you ready? Let's move on to simplifying and solving this equation to find the width of the rectangle.
Solving for the Width
Okay, team, let's roll up our sleeves and solve this equation! We've got 42 = 2(3w + 5) + 2w. The first thing we need to do is simplify the equation by getting rid of those parentheses. Remember the distributive property? We need to multiply the 2 by both terms inside the parentheses: 2 * 3w and 2 * 5. This gives us: 42 = 6w + 10 + 2w. Now, we can combine like terms. We have '6w' and '2w' on the right side of the equation, so let's add them together: 42 = 8w + 10. Great! We're one step closer. Now, we want to isolate the term with 'w' on one side of the equation. To do this, we need to get rid of the '+ 10'. We can do that by subtracting 10 from both sides of the equation: 42 - 10 = 8w + 10 - 10. This simplifies to: 32 = 8w. Almost there! Now, we just need to get 'w' by itself. It's being multiplied by 8, so we need to do the opposite operation – divide both sides by 8: 32 / 8 = 8w / 8. This gives us: 4 = w. Woohoo! We've found the width. The width of the rectangle is 4 cm. But hold on, we're not quite done yet. It's always a good idea to check our answer to make sure it makes sense. This is a crucial step in problem-solving. We've solved for 'w', but let's make sure it fits with the original problem. Why is this so important? Because it helps us avoid careless mistakes and build confidence in our solutions. Solving an equation is only half the battle; verifying the solution is the other half. So, before we declare victory, let's plug our answer back into the original problem and see if everything checks out.
Checking the Answer
Alright, detectives, let's verify our solution! We found that the width (w) of the rectangle is 4 cm. Now we need to make sure this answer makes sense in the context of the problem. Remember, the length (l) is three times the width plus 5 cm, so l = 3w + 5. Let's plug in our value for 'w': l = 3 * 4 + 5 l = 12 + 5 l = 17 cm. So, if the width is 4 cm, the length is 17 cm. Now, let's check the perimeter. The perimeter (P) should be 42 cm, and we know P = 2l + 2w. Let's plug in our values for 'l' and 'w': P = 2 * 17 + 2 * 4 P = 34 + 8 P = 42 cm. Awesome! Our calculated perimeter matches the given perimeter. This means our solution is correct. We've not only solved for the width but also verified that our answer fits the conditions of the problem. Why is this process of checking so vital? Because it's a powerful way to catch errors and ensure accuracy. In real-world scenarios, a mistake in calculations can have serious consequences, whether you're building a bridge, designing a machine, or managing your finances. By developing the habit of checking your work, you're building a valuable skill that will serve you well in many areas of life. So, we've successfully found the width of the rectangle and confirmed that our solution is correct. High five! But let's take a moment to reflect on the journey we've taken to get here. What key steps did we follow, and what valuable lessons did we learn along the way?
Conclusion
Alright, mathletes, we did it! We successfully navigated this rectangle problem from start to finish. We started by understanding the problem, translating the words into mathematical expressions, and setting up an equation. Then, we solved the equation, found the width, and, most importantly, we checked our answer. We proved that the width of the rectangle is indeed 4 cm. But beyond just getting the right answer, what have we learned in this process? We've reinforced the importance of breaking down complex problems into smaller, manageable steps. We've practiced using variables to represent unknowns and setting up algebraic equations. We've honed our skills in simplifying and solving equations. And we've emphasized the crucial role of checking our work to ensure accuracy. These are not just math skills; they are problem-solving skills that you can apply in countless situations, from planning a project to making important decisions. Math is not just about numbers and formulas; it's about developing logical thinking and problem-solving abilities. By tackling problems like this one, we're building our confidence and competence in math, and we're also equipping ourselves with tools that will help us succeed in other areas of life. So, keep practicing, keep asking questions, and keep exploring the wonderful world of mathematics. Who knows what exciting challenges we'll conquer next? Remember, every problem is an opportunity to learn and grow. And that’s a wrap, folks! Until next time, keep those math muscles flexed!
