Solving The Rectangular Garden Wire Problem

Hey guys! Let's dive into a fun math problem involving a rectangular garden, some poles, and wire. We're going to break it down step-by-step so it's super easy to understand. So, grab your coffee, get comfy, and let's get started. This problem is all about finding the dimensions of a rectangular garden given the total wire length used to fence it. The key to solving this type of problem is to carefully consider the information provided and break it down into manageable parts. We'll explore how to find the dimensions of the garden, which are whole numbers in meters, when the garden is fenced with three rows of wire, and the total wire length is 102 meters. This kind of problem often appears in math contests, entrance exams, and even real-world scenarios. We'll use basic algebraic principles and a bit of logical thinking to reach the solution. This is a great exercise for strengthening problem-solving skills and understanding how mathematical concepts apply to practical situations. Let's make sure we understand the core concepts and how they relate to the problem. We'll also cover useful tips and tricks to improve your problem-solving skills, and finally, look into some similar problems to sharpen your understanding. Let's unravel this math puzzle together!

Understanding the Problem: The Rectangular Garden and Wire

Let's begin by understanding the problem; We have a rectangular garden and poles placed at its corners. The garden is surrounded by three rows of wire, and the total length of the wire used is 102 meters. The crucial information here is that the lengths of the sides of the garden are whole numbers. Our main objective is to find the possible dimensions of the garden (length and width) in meters. The fact that the wire goes around the garden three times means that the total perimeter of the garden is a third of the total wire length. This simplifies our calculations, allowing us to focus on the perimeter rather than the entire wire length. We'll need to use this information, along with the fact that the sides are integers, to find the dimensions. The problem requires us to apply our knowledge of the perimeter of a rectangle, which is calculated as 2 times the sum of its length and width (2 * (length + width)). We also need to understand that the total wire length represents the total distance the wire covers around the garden, three times over. The concept of perimeter is fundamental to solving this problem. In a rectangle, the perimeter is the total distance around the outside. Think of it as walking around the garden; the perimeter is the total distance you would walk. Since the garden is surrounded by three rows of wire, the total wire length is actually three times the perimeter of the garden. By dividing the total wire length by three, we can calculate the perimeter of the garden. We'll need to use this perimeter to determine the possible values for the length and width of the garden. The condition that the lengths of the sides are whole numbers further refines the possible solutions. This constraint narrows down the potential combinations of length and width. With these concepts in mind, we can effectively approach the problem. It is designed to test your understanding of geometric shapes and mathematical concepts. These are very important in real life situations. We can apply this knowledge to other similar problems, making sure the learning process is continuous.

Setting Up the Equations: Perimeter and Wire Length

Now, let's convert the word problem into mathematical equations. The total length of the wire is 102 meters, and since the wire goes around the garden three times, the total perimeter of the garden is 102 meters divided by 3, which equals 34 meters. Mathematically, if we denote the length of the garden as 'L' and the width as 'W', the perimeter (P) of the garden can be represented as: P = 2 * (L + W). We know P = 34 meters. So, the equation becomes: 2 * (L + W) = 34. Simplifying this, we get: L + W = 17. The primary equation we have is L + W = 17, where L and W are integers, representing the length and width of the garden, respectively. This equation is the heart of our solution, as it relates the length and width of the rectangle. The challenge lies in finding integer values for L and W that satisfy this equation. The integer constraint is important here. It ensures that the lengths of the sides are whole numbers, as specified in the problem. The next step involves finding all possible pairs of integers (L, W) that add up to 17. Remember that both L and W must be positive, as a negative length or width doesn't make sense in this context. The equation L + W = 17 gives us several possible solutions for L and W. For example, L could be 1 and W could be 16, or L could be 2 and W could be 15, and so on. We should consider all the possible pairs of numbers that add up to 17. By listing all these pairs, we can identify all possible dimensions of the garden that satisfy the conditions of the problem. This is a crucial step in finding the solutions. This method of setting up equations and finding solutions is used frequently in various areas of mathematics and physics. Understanding how to model word problems mathematically is a core skill for solving problems. It's a foundational skill for various fields.

Finding the Possible Dimensions: Integer Solutions

Let's find all the possible integer solutions that satisfy the equation L + W = 17. Both L and W represent the length and width of the rectangle, and they must be positive whole numbers. We can systematically list all possible pairs of (L, W) as follows: (1, 16), (2, 15), (3, 14), (4, 13), (5, 12), (6, 11), (7, 10), (8, 9), (9, 8), (10, 7), (11, 6), (12, 5), (13, 4), (14, 3), (15, 2), (16, 1). Each of these pairs represents a possible combination of length and width for the rectangular garden. As you can see, the order of the numbers within each pair does not matter, because the garden is a rectangle. Therefore, (1, 16) is the same as (16, 1), and they represent the same dimensions. This means that we have several possible dimensions for the rectangular garden that satisfy the problem conditions. Each solution represents a valid set of dimensions. So, we've successfully determined all the possible combinations of length and width that meet the criteria. This step is about applying a clear methodology to identify all the feasible solutions. Recognizing and understanding integer solutions is a useful skill. The entire process requires a solid grasp of mathematical concepts and problem-solving strategies, ensuring a complete and accurate solution. This systematic approach guarantees we have found all the potential dimensions. These are very useful in real-world situations, showing that math is a practical tool for solving problems. The problem-solving strategy that we've employed is very versatile and applicable in many areas, including real-life scenarios. This is why learning this kind of problem is important for your development.

Verifying the Solutions: Ensuring Accuracy

Let's verify our solutions. It's always a good practice to double-check our work. Let's make sure that each pair of dimensions we found actually results in a perimeter of 34 meters. We can use the formula P = 2 * (L + W). For the pair (1, 16): P = 2 * (1 + 16) = 2 * 17 = 34 meters. For the pair (2, 15): P = 2 * (2 + 15) = 2 * 17 = 34 meters. We can see that all the pairs listed in the previous step, when plugged into the perimeter formula, will indeed result in a perimeter of 34 meters. Each solution we derived satisfies the initial condition. Our answers are consistent with the original problem statement. This verification step confirms the accuracy of our approach and results. It's essential to confirm that our solutions match all the given requirements, which in this case, include the integer side lengths and the total wire length. This is an important part of the problem solving process, confirming that the solutions are correct. In this case, we have a way to easily check. In other scenarios, you might use a more complex method of verification. So, through this, we have effectively confirmed our calculations. We have successfully verified our solutions, ensuring their accuracy. This step of verification enhances our confidence in the results and confirms our understanding of the problem. This practice is crucial in any mathematical problem solving, ensuring reliability. It also solidifies your skills in this area. Through such exercises, we can improve our problem-solving skills, and apply these methods to any problem.

Conclusion: Summary of the Solution

To recap, here's what we've done. We started with a word problem about a rectangular garden surrounded by wire. We were given that the total wire length was 102 meters, and the wire was wrapped around the garden three times. We then determined the perimeter of the garden by dividing the total wire length by 3, resulting in a perimeter of 34 meters. We then set up the equation L + W = 17, where L is the length and W is the width of the garden. After that, we found all the possible integer pairs of L and W that satisfy this equation, ensuring that both lengths are positive whole numbers. We verified each solution to confirm that the calculated perimeter matched the original problem conditions. The possible dimensions of the rectangular garden are: (1, 16), (2, 15), (3, 14), (4, 13), (5, 12), (6, 11), (7, 10), (8, 9), (9, 8), (10, 7), (11, 6), (12, 5), (13, 4), (14, 3), (15, 2), and (16, 1). This is all the solutions we can provide. We've used a combination of problem-solving techniques, ensuring that we approached it systematically and accurately. Congratulations, you've solved this problem! It shows how math concepts, like perimeter and integer solutions, are used in practical problems. We can apply this approach to similar problems. This exercise demonstrates how mathematics is a helpful tool for problem-solving. This problem helps in reinforcing basic mathematical principles. This will help you in further mathematical challenges.

Tips for Similar Problems: Key Strategies

Here are a few tips to help you tackle similar problems. First, always read the problem carefully and understand what's being asked. Identify the key information and constraints. Drawing a diagram can be incredibly helpful for visualizing the problem, especially in geometry problems like this one. Next, break down the problem into smaller, manageable steps. This makes it easier to tackle the problem systematically. Translate the word problem into mathematical equations. This is often the most critical step. Ensure that you correctly use the given information. Don't forget to consider all the constraints. In our case, this was the integer requirement for the lengths and widths. Remember to double-check your work. This helps to catch any mistakes. Practice makes perfect. The more problems you solve, the better you'll become. By using these strategies, you'll be well-equipped to face similar challenges with confidence. Keep practicing and you will get better. These tips will greatly improve your problem-solving skills. Remember that practice is key to mastering these techniques. These are useful tips in many areas of life. Use these tips to hone your problem-solving skills. Make sure you understand all the techniques. And finally, stay curious. Keep exploring new math concepts and challenges.

Further Practice: Related Problems

Let's try some related problems to help solidify your understanding. Here are some examples to practice with. Problem 1: A rectangular field is to be fenced using 40 meters of wire. If the lengths of the sides are whole numbers, what are the possible dimensions of the field? Problem 2: A rectangular room has a perimeter of 28 meters. If the length and width of the room are whole numbers, find all possible combinations of length and width. Problem 3: The perimeter of a rectangle is 50 meters, and the length and width are integers. If the length is more than twice the width, find the possible dimensions of the rectangle. These problems are designed to test your understanding of perimeter and integer solutions. You can apply the same problem-solving techniques we discussed. Remember to follow the steps we've laid out: understand the problem, set up equations, find solutions, and verify your answers. Solving these problems will help you reinforce what you've learned. By tackling these problems, you'll gain confidence and sharpen your problem-solving skills. These extra problems will help reinforce your understanding of the concepts. These problems are designed to enhance your comprehension. Don't hesitate to give them a try. They are designed to improve your skills. Practice these questions and you'll find it easier to tackle these types of questions. Take your time, and remember that practice makes perfect!

Hope this helps, guys! Keep practicing, and you'll become a math pro in no time! Keep exploring and having fun with math! Bye!