Find The Length Of A Rectangle: A Math Problem

Hey guys! Let's dive into a fun math problem. We're going to figure out the length of a rectangle. The area of a rectangle is a fundamental concept in geometry, and understanding how to calculate its dimensions is super important. The area of a rectangle is given as 9.312 x 10^4 cm², and the width is 2.56 x 10^2 cm. Our mission, should we choose to accept it, is to find the length. This isn't some super-complex equation – it's just a matter of understanding the relationship between area, length, and width, and then applying the right formula.

Let's break down the concepts before we jump into the problem-solving. The area of a rectangle is the total space inside the rectangle. It's calculated by multiplying the length by the width. So, if you know the area and one side (either the length or the width), you can always find the other side. In this case, we're given the area and the width, so we need to find the length. This is a perfect opportunity to show you how to manipulate the basic formula and solve for an unknown variable. The use of scientific notation might look a little intimidating at first, but trust me, we can break it down. Scientific notation is just a handy way to write very large or very small numbers. Once you get the hang of it, it's a piece of cake.

Remember that, the area of a rectangle is found using this formula: Area = Length × Width. Now, to find the length, we need to rearrange this formula. We can do this by dividing both sides of the equation by the width. So, the formula we'll use is: Length = Area / Width. That's right, we're using basic algebra here to solve for the unknown. We'll take the given area, divide it by the given width, and then the result will be the length. Easy peasy, right? With a little practice, you'll be able to solve a lot of these types of problems without breaking a sweat. So grab your calculators, or even just a piece of paper and a pencil and let's get started. I promise, it's not as hard as it looks. The key is to stay organized and take it step by step. And if you get stuck, don't worry about it, we'll figure it out together!

The Formula and the Calculation

Alright, guys, let's get down to business and calculate the length of that rectangle! As we discussed before, we're going to use the formula: Length = Area / Width. Now, we have the area of the rectangle which is 9.312 x 10^4 cm². Also, we have the width, which is 2.56 x 10^2 cm. Let's substitute those values into our formula. So, Length = (9.312 x 10^4) / (2.56 x 10^2). Now, before we start punching numbers into a calculator, let's make things a little easier on ourselves.

We can handle this calculation in a few different ways. Some of us can use a calculator. Most calculators can handle scientific notation directly, and you can input the numbers as they are. However, for those of you who aren't super familiar with scientific notation, or maybe just like to break things down step by step, here’s another approach: First, we can divide the numbers (9.312 / 2.56). This gets us 3.6375. Next, we need to deal with the powers of 10. When dividing numbers in scientific notation, we subtract the exponents. So, we have 10^4 / 10^2, which is the same as 10^(4-2), that's 10^2. So the answer is 3.6375 x 10^2. Now, let's recap what we did: We divided the coefficient (the number in front of the “x 10”) and then we managed the exponents. Once we've got these two components, we can assemble our final answer! Make sure to always include units. In this case, since we're talking about length, our units are centimeters (cm). So the length of the rectangle is 3.6375 x 10^2 cm. Easy enough, right? I told you it wasn't so bad.

Diving into the Answer Choices

Now, let's take a look at the multiple-choice options given and find out which one matches our answer. We calculated that the length is 3.6375 x 10^2 cm. If you look at the options given, you'll quickly see which one is the correct choice. The first option, A, says 3.6375 x 10^2. This is exactly what we calculated. So, A is the correct answer. Awesome, we solved the problem! Let's check the other choices to make sure we're on the right track.

Option B says 3.6375 x 10^6. This answer is way off because the exponent is much larger than what we calculated, so we can discard it. The exponent being so large suggests that the final result would be a value in the millions, which wouldn't make sense for a rectangle with an area in the ten-thousands and a width in the hundreds. This option shows how easy it is to make a mistake with scientific notation if you're not careful. Option C says 6.752 x 10^2. This is also incorrect, the coefficient (the number in front of the x 10) does not match our correct answer. Finally, option D says 6.752. We can eliminate this one right away because it doesn't even include the scientific notation. We can see that each incorrect option has a significant error either in the coefficient or the exponent, which highlights the importance of paying close attention to all parts of the equation. So, we've confirmed that our answer, 3.6375 x 10^2, is indeed the right one. Nice work everyone!

Why This Matters

Why does all this matter? Well, understanding how to calculate the area and dimensions of rectangles is useful in everyday life. From figuring out how much paint you need to cover a wall to determining how much space you have for furniture in your room. It is crucial for various fields, like architecture, engineering, and even interior design. Imagine designing a building or a room; you will need to know how to work with areas and dimensions. Being able to quickly and accurately perform these calculations can save you time and reduce the risk of making mistakes.

Also, knowing this skill can give you a better understanding of the world around you. Math isn't just about numbers; it's about the ability to understand and solve problems. It trains your mind to think logically, which is a valuable asset in any field. Whether you’re planning a garden, building a bookshelf, or just trying to figure out if your new sofa will fit in the living room, knowing how to calculate area and dimensions is a super practical skill to have. Plus, as you become more comfortable with these calculations, you'll build your confidence. You'll feel a sense of accomplishment when you solve a problem, and that sense of accomplishment will motivate you to learn more. So, keep practicing, keep asking questions, and don't be afraid to make mistakes. Every mistake is an opportunity to learn and grow. Keep it up, guys, and you will be great!