Solving Rectangle Area: A Step-by-Step Guide

Hey there, math enthusiasts! Let's dive into a classic geometry problem. This one is all about rectangles, perimeters, areas, and a little bit of algebraic thinking. So, grab your pencils, and let's get started. We're going to break down how to find the area of a rectangle when you're given its perimeter and a relationship between its base and height. In this case, we have a rectangle with a perimeter of 60 meters. Also, the base of the rectangle is twice the size of its height. Now, our goal is to figure out the area of this rectangle. We'll go through the problem step-by-step, making sure everything is super clear and easy to follow. If you are having problems in math, this is the right place to be. We will work to solve the problem by doing math and applying it to real-world examples. Are you ready?

Understanding the Problem: The Foundation for Success

First things first, let's make sure we totally grasp what the question is asking. We've got a rectangle, which, as you probably know, is a four-sided shape with four right angles. The opposite sides of a rectangle are equal in length. Now, the perimeter is the total distance around the outside of the rectangle—imagine walking all the way around the edges; that's the perimeter. The area, on the other hand, is the space inside the rectangle. Think of it as the amount of carpet you'd need to cover the floor. In this problem, we're given the perimeter (60 meters) and a relationship between the base and height: the base is twice the height. We want to find the area. The trick here is to use the information about the perimeter to find the dimensions (length and width, or base and height) of the rectangle, and then use those dimensions to calculate the area. It is important to remember the formulas for perimeter and area; they are the keys to unlocking this problem. Knowing these formulas is the foundation on which we will build our solution. So let's review them quickly. The perimeter of a rectangle is calculated as: P = 2 * (base + height), where P represents the perimeter. The area of a rectangle is calculated as: A = base * height, where A represents the area. Keep these handy—they're our secret weapons!

Now, let's translate the information from the problem into mathematical terms. We know the perimeter (P) is 60 m. Let's represent the height as 'h' and the base as 'b'. The problem tells us that the base is twice the height, so we can write this as: b = 2h. This is a crucial piece of information. Since we have two unknowns (base and height), we need more information to solve the problem. The perimeter equation will help us create another equation with these unknowns. Using the formula for the perimeter of a rectangle, which is P = 2(b + h). Remember that we know P = 60 m and b = 2h. Now, let's substitute the values we know into the formula: 60 = 2(2h + h). Let's simplify this equation to find the value of h.

Finding the Dimensions: Unveiling the Base and Height

Alright, folks, time to get our hands dirty with some algebra! We've got our perimeter equation: 60 = 2(2h + h). Our goal here is to isolate 'h' to find the height of the rectangle. Step 1: Simplify inside the parentheses. Inside the parentheses, we have 2h + h, which simplifies to 3h. Now our equation looks like this: 60 = 2(3h). Step 2: Multiply. Multiply 2 by 3h. This gives us: 60 = 6h. Step 3: Solve for 'h.' To isolate 'h,' we need to divide both sides of the equation by 6. This gives us: h = 10. So, we now know that the height (h) of the rectangle is 10 meters. Cool, right? But we're not done yet; we still need to find the base. Remember, the base (b) is twice the height (h). We know h = 10 m, so: b = 2 * 10 = 20 m. We've found the base! The base of the rectangle is 20 meters. Now, with the height and base known, we can finally calculate the area. We have the dimensions of our rectangle. The height is 10 meters, and the base is 20 meters. We're on the home stretch!

Calculating the Area: The Grand Finale

Okay, team, we've got the dimensions: height = 10 meters, base = 20 meters. Now it's time to find the area. The area of a rectangle is calculated using the formula: Area = base * height. We know the base (20 m) and the height (10 m), so let's plug those values into the formula: Area = 20 m * 10 m. Now, perform the multiplication: 20 * 10 = 200. Therefore, the area of the rectangle is 200 square meters (m²). And there you have it, folks! We've solved the problem. By breaking down the problem step-by-step, using the right formulas, and a little bit of algebra, we were able to find the area of the rectangle. Isn't math amazing? From start to finish, we've transformed a word problem into a concrete solution. We've gone from the perimeter to finding the height and base and, finally, to calculating the area. You see, with the right approach and a little practice, math problems like these become manageable and even fun. Now we can confidently say that the correct answer to the question is (A) 200. That's the power of understanding and applying the concepts. Feel proud of your success, guys!

Conclusion: Mastering the Rectangle

So there you have it! We've successfully navigated the world of rectangles, perimeters, and areas. We started with a word problem, broke it down into smaller, manageable parts, and used the power of formulas and algebra to find the solution. Remember, the key takeaways from this problem are: Understand the formulas for perimeter and area of a rectangle. Know how to use the relationship between the base and height (or any other dimensions). Practice using algebraic techniques to solve for unknowns. The more you practice these types of problems, the easier they will become. You'll start to recognize patterns and develop your problem-solving skills. Math isn't about memorization; it's about understanding the concepts and applying them creatively. The cool part is that these skills translate beyond just math class. Problem-solving is a valuable skill in every area of life. So, keep practicing, keep learning, and keep asking questions. If you found this explanation helpful, give it a thumbs up, share it with your friends, and subscribe for more math tutorials. Thanks for joining me on this mathematical journey! Until next time, happy calculating, and keep exploring the amazing world of mathematics!