Rectangle Dimensions: Solving Perimeter Problems

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Rectangle Dimensions: Solving Perimeter Problems

Hey math enthusiasts! Let's dive into a classic geometry problem. We're going to figure out the length and width of a rectangle. Here's the deal: the length of this rectangle is three times its width. Plus, we know the perimeter is 72 meters. So, how do we find those dimensions? Don't worry, it's easier than you might think. This problem is a great example of how algebra and geometry work together, and it's super practical because we can apply this knowledge to lots of real-world situations, like figuring out how much fencing we need for a rectangular yard, or the dimensions of a picture frame. The key is to break down the problem into smaller, manageable steps. We'll translate the words into mathematical expressions, set up an equation, and then solve for the unknown variables. Sounds good, right? Let's get started and unravel this geometry puzzle together. This problem-solving approach is not just about getting an answer; it's about building a strong foundation in math, helping you tackle more complex problems with confidence. It's about seeing the beauty and practicality of mathematics in action. It's like learning a new language – once you get the hang of it, you can communicate and solve problems in a whole new way. Ready to unlock the secrets of this rectangle? Let's go!

Setting Up the Problem: Translating Words into Equations

Alright guys, let's get down to the nitty-gritty and translate our word problem into math speak. We have a rectangle, and we need to find its length and width. Here's what we know:

  1. The length is three times the width: This is a crucial piece of information. We can express this mathematically. Let's use 'l' to represent the length and 'w' to represent the width. So, we can write this as: l = 3w. This equation tells us that the length is equal to three times the width. If the width is 2 meters, the length is 6 meters, and so on. Pretty straightforward, huh?
  2. The perimeter is 72 meters: The perimeter is the total distance around the rectangle. For any rectangle, the perimeter is calculated by adding up the lengths of all four sides. Since a rectangle has two lengths and two widths, the formula for the perimeter (P) is: P = 2l + 2w. We know the perimeter is 72 meters, so we can write: 72 = 2l + 2w. Now we have two equations:
    • l = 3w
    • 72 = 2l + 2w

These two equations are the keys to solving the problem. The first one gives us the relationship between length and width, and the second one gives us the total perimeter. Our next step is to use these equations to find the values of 'l' and 'w'. Think of it like this: we're building a bridge from the information we have to the answers we need. Each equation is a supporting pillar, and together, they help us cross over to our solution. This is where the real fun begins! We'll use a method called substitution to solve for the unknowns. Trust me, it's not as scary as it sounds. It's all about plugging one equation into another to simplify things and get our answers. Ready to keep going?

Solving for Width and Length: Using Substitution

Okay, team, time to put on our solving hats! We've got our equations, and now we're going to find the width and length. We're going to use a method called substitution. This is where we use one equation to replace a variable in another equation. Here's how it works:

  1. Substitute l = 3w into 72 = 2l + 2w: Since we know that l is equal to 3w, we can replace 'l' in the second equation with 3w. This gives us: 72 = 2(3w) + 2w. See what we did there? We just plugged in 3w for l.
  2. Simplify the equation: Now, let's simplify the new equation. 2(3w) is the same as 6w, so our equation becomes: 72 = 6w + 2w. Combining like terms (6w and 2w), we get: 72 = 8w.
  3. Solve for w: To find the value of w, we need to isolate it. We can do this by dividing both sides of the equation by 8: 72 / 8 = 8w / 8. This gives us: 9 = w. So, the width (w) of the rectangle is 9 meters! We're making great progress, guys.

Now, we have the width, but we still need the length. Remember our first equation, l = 3w? We can use this to find the length. We know that w = 9, so we substitute this value into the equation: l = 3 * 9. This gives us: l = 27. So, the length (l) of the rectangle is 27 meters. We did it! We found both the width and the length. This substitution method is a powerful tool. It allows us to solve systems of equations by expressing one variable in terms of another. It's like having a secret code that unlocks the answers. Remember, practice makes perfect, and the more you practice, the more comfortable you'll become with this method. It's not just about solving this particular problem; it's about building a strong foundation for tackling more complex math problems in the future. We're equipping ourselves with the skills to confidently approach any challenge that comes our way. That is the goal of our learning journey!

Checking Your Work: Does it Make Sense?

Alright, before we high-five ourselves, let's do a quick check to make sure our answers make sense. It's always a good idea to verify your work, so we don't make silly mistakes. This also helps to build confidence in our problem-solving skills, and helps us catch any errors. Here’s how we can check our answers:

  1. Check the relationship between length and width: We found that the length is 27 meters and the width is 9 meters. According to the problem, the length should be three times the width. Let's see if that's true: 27 = 3 * 9. Yep, it checks out! Our length is indeed three times our width.
  2. Calculate the perimeter: The problem tells us that the perimeter should be 72 meters. Let's calculate the perimeter using our values for length and width. Remember, the perimeter is 2l + 2w. So, we have: 2 * 27 + 2 * 9 = 54 + 18 = 72. Bingo! Our calculated perimeter matches the given perimeter. This confirms that our answers are correct. Always take a moment to double-check your work. It's a critical step in problem-solving. It's like proofreading a paper or reviewing code before running it. It saves us from potential errors and reinforces our understanding of the concepts. Also, it’s a great way to make sure we've properly understood the problem and applied the correct formulas and techniques. By checking our work, we not only ensure the accuracy of our solutions but also boost our confidence in our ability to solve problems effectively. That's the beauty of math – the ability to verify our solutions and build trust in our capabilities. Keep in mind that mistakes are a natural part of the learning process. But by checking our work, we can identify and learn from our mistakes, making us stronger mathematicians. So, give yourselves a pat on the back; we've successfully solved the problem and verified our answers!

Conclusion: Mastering Rectangle Dimensions

Fantastic job, everyone! We've successfully found the length and width of the rectangle, and we've verified our answers. Here's a quick recap of what we've accomplished:

  • We understood the problem, identifying the given information (length is three times the width, and the perimeter is 72 meters). We then translated the problem into mathematical equations: l = 3w and 72 = 2l + 2w.
  • We used the substitution method to solve the equations. We substituted l = 3w into the perimeter equation and simplified to find the width (w = 9 meters).
  • We then used the relationship l = 3w to find the length (l = 27 meters).
  • Finally, we checked our work to ensure our answers were accurate, verifying that the length was three times the width and that the calculated perimeter matched the given perimeter. This problem illustrates how basic geometry and algebra work hand in hand. By understanding these concepts and practicing the methods, you can confidently tackle similar problems. The ability to translate real-world scenarios into mathematical equations and then solve them is a valuable skill that has applications across many fields. Keep practicing, and you'll find that these kinds of problems become easier and more enjoyable. Remember, every problem you solve is a step forward in your mathematical journey. So, keep exploring, keep learning, and keep the curiosity alive. The world of mathematics is vast and exciting, and there is always something new to discover. You're now equipped with the tools to solve this specific problem and, more importantly, to approach similar problems with confidence. Well done, guys! You’ve shown great teamwork today. Keep up the excellent work, and always remember the joy of solving a problem. Feel free to explore more problems, and keep the mathematical spirit burning bright! Until next time, keep calculating and keep having fun with math.