Finding The Maximum Width Of A Rectangle Given Perimeter And Length

Hey guys! Let's dive into a super practical math problem today – figuring out the maximum width of a rectangle when we know its perimeter can't be more than 150 centimeters and its length is a solid 40 centimeters. Sounds like fun, right? This kind of problem pops up in all sorts of real-life scenarios, from planning room layouts to figuring out material needs for a project. So, let's break it down step by step and get to the bottom of this. We'll cover the basics of rectangle perimeters, set up an inequality to represent the problem, and then solve for the maximum width. Trust me, it’s easier than it sounds, and by the end, you’ll be a pro at tackling these kinds of questions. Let's get started and see how math helps us in everyday situations!

Understanding the Basics of Rectangle Perimeter

Alright, so before we jump into solving this specific problem, let’s make sure we're all on the same page about what perimeter actually means and how it applies to a rectangle. Think of the perimeter as the total distance around the outside of a shape. Imagine you're building a fence around your backyard – the perimeter is the total length of fencing you'll need. For a rectangle, which has two pairs of equal sides (length and width), the perimeter is calculated by adding up the lengths of all four sides. This leads us to a neat little formula that makes things super easy.

The formula for the perimeter (P) of a rectangle is P = 2l + 2w, where 'l' stands for the length and 'w' represents the width. This formula basically says, "Take the length, double it, then take the width, double it, and add those two results together." It’s a straightforward way to find out how much distance you'd cover if you walked all the way around the rectangle. Now, why is this formula so important? Well, it’s the key to solving a ton of problems involving rectangles, like our main question about the maximum width. By understanding this basic concept, we can easily plug in the values we know and figure out what we're missing. So, with this formula in our toolkit, we're ready to tackle the problem head-on. Remember, math isn't just about numbers; it's about understanding the relationships between them. And in this case, understanding the relationship between a rectangle's length, width, and perimeter is going to save the day!

Setting Up the Inequality

Okay, now that we've got the basics of rectangle perimeters down, let's get to the heart of our problem. We know the perimeter of our rectangle can be no greater than 150 centimeters, and the length is set at 40 centimeters. The question we're trying to answer is: what's the maximum width this rectangle can be? To solve this, we're going to translate these words into a mathematical inequality. Think of an inequality as a way of showing a range of possibilities rather than one exact answer. In this case, our perimeter isn't just one number; it's a range of numbers up to 150 centimeters.

First, let's rewrite the perimeter formula with the information we have. We know P = 2l + 2w, and we're given that l = 40 centimeters. So, we can substitute that in: P = 2(40) + 2w, which simplifies to P = 80 + 2w. Now, here's the crucial part: the problem states the perimeter is “no greater than” 150 centimeters. This means the perimeter can be 150 centimeters or less, but not more. Mathematically, we express “no greater than” using the less than or equal to symbol (≤). So, we can write our inequality as: 80 + 2w ≤ 150. This inequality is the key to unlocking our solution. It tells us that 80 plus twice the width must be less than or equal to 150. We've successfully turned a word problem into a math problem! Now, the next step is to solve this inequality to find the maximum possible value for the width (w). Inequalities might seem a bit scary at first, but they're just like equations with a little extra flexibility. Stick with me, and we'll crack this!

Solving for the Maximum Width

Alright, guys, we've reached the exciting part where we actually solve for the maximum width! We've got our inequality set up: 80 + 2w ≤ 150. Now, it's time to put on our detective hats and figure out what 'w' can be. Solving an inequality is a lot like solving an equation, with one important twist that we'll talk about later. Our goal is to isolate 'w' on one side of the inequality, just like we would when solving for 'x' in an equation.

First, let's get rid of that 80 that's hanging out with the 2w. We do this by subtracting 80 from both sides of the inequality. Remember, whatever we do to one side, we have to do to the other to keep things balanced. So, we have: 80 + 2w - 80 ≤ 150 - 80, which simplifies to 2w ≤ 70. Awesome! We're one step closer. Now, we need to get 'w' all by itself. It's currently being multiplied by 2, so to undo that, we'll divide both sides of the inequality by 2. This gives us: (2w) / 2 ≤ 70 / 2, which simplifies to w ≤ 35. Boom! We've solved it! This inequality, w ≤ 35, tells us that the width of the rectangle must be less than or equal to 35 centimeters. So, what's the maximum width? Well, it's the highest value that 'w' can be while still satisfying the inequality. In this case, the maximum width is 35 centimeters. How cool is that? We took a word problem, turned it into an inequality, and solved it to find our answer. Remember, the key to solving inequalities is to treat them like equations, using inverse operations to isolate the variable. And now you've got another tool in your math belt! Let’s keep going and see how we can apply this knowledge.

Checking the Solution

Fantastic job, everyone! We've calculated that the maximum width of the rectangle is 35 centimeters. But before we declare victory, it's always a good idea to double-check our work. This step is super important because it helps us catch any mistakes and ensures our answer makes sense in the context of the original problem. So, how do we check our solution? We're going to plug the maximum width we found (35 centimeters) back into the perimeter formula and see if the resulting perimeter is indeed no greater than 150 centimeters.

Remember our perimeter formula: P = 2l + 2w. We know the length (l) is 40 centimeters, and we've calculated the maximum width (w) to be 35 centimeters. Let's plug those values in: P = 2(40) + 2(35). Now, let's do the math. 2 times 40 is 80, and 2 times 35 is 70. So, our equation becomes: P = 80 + 70, which simplifies to P = 150 centimeters. Bingo! The perimeter is exactly 150 centimeters, which meets the condition “no greater than 150 centimeters.” This confirms that our solution is correct. But checking our work isn’t just about plugging numbers into a formula. It’s also about thinking logically about the answer. Does it make sense that the maximum width is 35 centimeters, given the length and the perimeter constraint? In this case, it does. If the width were any larger, the perimeter would exceed 150 centimeters. By taking the time to check our solution, we've not only verified our math but also strengthened our understanding of the problem. So, always remember to double-check – it's a key ingredient in becoming a math whiz!

Real-World Applications

Okay, so we've successfully figured out the maximum width of a rectangle given certain constraints. But you might be wondering, "Where would I ever use this in real life?" Well, the truth is, these kinds of problems pop up in all sorts of situations! Understanding how to calculate perimeters and optimize dimensions can be incredibly useful in various fields and everyday scenarios. Let’s explore some cool real-world applications.

Imagine you're planning a garden and want to build a fence around it. You have a limited amount of fencing material, say 150 feet, and you want the garden to be 40 feet long. You need to figure out the maximum width you can make the garden without running out of fencing. Sounds familiar, right? This is exactly the kind of problem we just solved! Or, think about interior design. You might be arranging furniture in a room and want to ensure you have enough space to move around comfortably. Knowing the perimeter of a room and the dimensions of your furniture can help you optimize the layout. Architects and engineers use these concepts all the time when designing buildings and structures. They need to calculate perimeters and areas to ensure they're using materials efficiently and meeting safety requirements. Even in manufacturing, businesses need to optimize the dimensions of packaging to minimize material costs and shipping expenses. These are just a few examples, but the point is, the ability to work with perimeters, areas, and inequalities is a valuable skill in many different contexts. By mastering these concepts, you're not just learning math; you're equipping yourself with problem-solving tools that can be applied in the real world. So, keep practicing, keep exploring, and you'll be amazed at how useful math can be!

Conclusion

Alright, we've reached the end of our mathematical journey, and what a journey it has been! We started with a seemingly simple question about the maximum width of a rectangle and ended up exploring the concepts of perimeters, inequalities, and real-world applications. How awesome is that? We learned that the perimeter of a rectangle is the total distance around its outside, and we can calculate it using the formula P = 2l + 2w. We also tackled the challenge of translating a word problem into a mathematical inequality, which is a crucial skill in problem-solving. By setting up and solving the inequality, we found that the maximum width of our rectangle was 35 centimeters. And, importantly, we checked our solution to make sure it made sense.

But perhaps the most valuable takeaway is recognizing how these mathematical concepts connect to the real world. From planning gardens to designing buildings, understanding perimeters and dimensions is essential in countless situations. So, what's the big picture here? Math isn't just about memorizing formulas and solving equations; it's about developing critical thinking skills and learning how to approach problems systematically. By breaking down complex problems into smaller steps, we can find solutions and gain a deeper understanding of the world around us. Keep practicing, stay curious, and never stop exploring the amazing world of mathematics. You've got this!