Rectangle Length: Area & Width Given - Easy Solution

Hey guys! Today, we're diving into a super interesting math problem that involves finding the length of a rectangle when we know its area and width. Sounds like a puzzle, right? Well, let's grab our detective hats and solve it together! We'll break down each step, making sure it's crystal clear and maybe even a little fun. So, buckle up, and let's get started!

Setting the Stage: Understanding the Problem

Before we jump into calculations, let's make sure we understand what we're dealing with. We have a rectangle, and we know two crucial things about it: its area and its width. The area, which is the space inside the rectangle, is given by the expression 45x² - 42x - 48. Think of 'x' as a mystery number we don't know yet, but it's part of the rectangle's dimensions. The width, which is one of the sides of the rectangle, is 5x - 8. Our mission, should we choose to accept it (spoiler: we do!), is to find the length of the other side of the rectangle. Remember that the area of a rectangle is always found by multiplying its length and width. So, if we know the area and the width, we can find the length by doing the opposite of multiplication, which is division! This means we need to divide the area (45x² - 42x - 48) by the width (5x - 8). This division isn't as scary as it looks, trust me. We'll use a technique called polynomial long division to solve it. It's similar to the long division you learned with numbers, but with algebraic expressions. By understanding the relationship between area, length, and width, we're setting a solid foundation for tackling the problem head-on. So, let's roll up our sleeves and get ready to divide!

Diving into Polynomial Long Division

Okay, guys, here comes the main event: polynomial long division! If you've done regular long division with numbers, this will feel familiar, just with a bit of algebra sprinkled in. Our goal is to divide the polynomial 45x² - 42x - 48 by 5x - 8. First, set up the long division. Write 45x² - 42x - 48 inside the division symbol (this is our dividend) and 5x - 8 outside (this is our divisor). Now, focus on the first terms of both the dividend and the divisor. Ask yourself: "What do I need to multiply 5x by to get 45x²?" The answer is 9x. Write 9x above the division symbol, aligned with the -42x term. Next, multiply the entire divisor (5x - 8) by 9x: 9x * (5x - 8) = 45x² - 72x. Write this result below the dividend, aligning like terms. Subtract the result from the dividend: (45x² - 42x - 48) - (45x² - 72x) = 30x - 48. Bring down the next term from the dividend (which is -48), so now we have 30x - 48. Repeat the process. Ask yourself: "What do I need to multiply 5x by to get 30x?" The answer is 6. Write +6 above the division symbol, aligned with the -48 term. Multiply the divisor (5x - 8) by 6: 6 * (5x - 8) = 30x - 48. Write this result below 30x - 48. Subtract: (30x - 48) - (30x - 48) = 0. Since the remainder is 0, we're done! The quotient, which is the expression above the division symbol, is 9x + 6. This is the length of the rectangle!

The Grand Finale: Expressing the Length

Alright, guys, after all that division, we've arrived at our answer! The length of the rectangle is 9x + 6. But, as any good mathematician knows, it's always a good idea to double-check our work. To do this, we can multiply the length we found (9x + 6) by the width we were given (5x - 8) and see if we get back the area (45x² - 42x - 48). Let's do it: (9x + 6) * (5x - 8) = 45x² - 72x + 30x - 48 = 45x² - 42x - 48. Woo-hoo! It matches the area we started with, so we know we did the division correctly. Therefore, we can confidently say that the length of the rectangle is indeed 9x + 6. This result tells us how the length of the rectangle changes depending on the value of 'x'. If 'x' is a large number, the length will also be large, and if 'x' is a smaller number, the length will be smaller too. So, there you have it! We've successfully found the length of the rectangle using polynomial long division. Give yourselves a pat on the back; you've earned it!

Real-World Connections: Why This Matters

Now, you might be thinking, "Okay, this is cool, but when am I ever going to use this in real life?" Well, guys, understanding how algebraic expressions and geometric shapes relate to each other is super useful in many fields. For example, architects and engineers use these kinds of calculations all the time when designing buildings and structures. They need to know how the dimensions of different parts of a building affect its overall area and volume. Similarly, in manufacturing, companies use these concepts to optimize the use of materials and reduce waste. Understanding how to manipulate algebraic expressions also helps in computer graphics and game development. When creating 3D models, developers need to calculate the sizes and shapes of objects accurately. Even in finance, these concepts can be applied to understand growth rates and compound interest. By mastering the basics of algebra and geometry, you're equipping yourself with valuable tools that can be applied in a wide range of real-world situations. So, the next time you see a building or play a video game, remember that the concepts we learned today played a part in making it happen!

Wrapping Up: Key Takeaways

Alright, guys, let's wrap up what we've learned today. We started with a rectangle, knowing its area (45x² - 42x - 48) and width (5x - 8), and our mission was to find its length. To do this, we used a technique called polynomial long division, which is similar to regular long division but with algebraic expressions. We divided the area by the width and found that the length of the rectangle is 9x + 6. Remember, the key to solving this problem was understanding the relationship between area, length, and width, and applying the correct mathematical operation (division) to find the missing dimension. We also talked about why these concepts are important in the real world, from architecture and engineering to manufacturing and finance. By mastering these skills, you're not just learning math; you're developing problem-solving abilities that will serve you well in many aspects of life. So, keep practicing, keep exploring, and never stop asking "why?" You guys are awesome, and I'm excited to see what you'll accomplish next!