Determining The Width Of A Concrete Path Around A Rectangular Field

Introduction

In this article, we will delve into a classic geometry problem involving a rectangular field surrounded by a concrete path of uniform width. We'll explore the concepts of area, dimensions, and algebraic equations to determine the width of the path. This type of problem is not only a great exercise in mathematical reasoning but also has practical applications in real-world scenarios such as landscaping and construction. Specifically, we aim to solve the following problem: A rectangular field, 70 m long and 50 m wide, is surrounded by a concrete path of uniform width. Given that the area of the path is $1024 m^2$, find the width of the path. This problem requires us to translate a geometric situation into an algebraic equation and then solve it. Let's embark on this mathematical journey together!

Problem Statement and Setup

Let's start by clearly stating the problem we intend to solve. We have a rectangular field with a length of 70 meters and a width of 50 meters. This field is surrounded by a concrete path of uniform width. The area of this path is given as 1024 square meters. Our goal is to find the width of this path. This involves visualizing the situation, identifying the relevant geometric shapes, and setting up an equation that relates the known quantities to the unknown width. A clear understanding of the problem statement is crucial for formulating an effective solution strategy.

To set up the problem, let's denote the width of the path by x meters. This is the key variable we need to determine. The outer dimensions of the rectangle formed by the field and the path will then be (70 + 2x) meters in length and (50 + 2x) meters in width. This is because the path extends x meters on each side of the field. The total area enclosed by the outer rectangle (field + path) can be expressed as the product of its length and width, which is (70 + 2x)(50 + 2x) square meters. The area of the field itself is simply 70 * 50 = 3500 square meters. The area of the path is the difference between the total area and the area of the field, which leads us to the equation (70 + 2x)(50 + 2x) - 3500 = 1024. This equation will be the foundation for our solution.

Forming the Equation

Now, let's translate the problem into a mathematical equation. As established earlier, the area of the path is the difference between the area of the outer rectangle (field + path) and the area of the field itself. We have the area of the path as 1024 square meters, the dimensions of the field as 70 m by 50 m, and we've denoted the width of the path as x meters. The outer rectangle's dimensions are (70 + 2x) meters and (50 + 2x) meters. Therefore, the area of the outer rectangle is (70 + 2x)(50 + 2x) square meters. This process of converting the geometric information into an algebraic form is a fundamental step in solving mathematical problems.

The equation representing the given situation can be written as follows: (70 + 2x)(50 + 2x) - 3500 = 1024. This equation captures the relationship between the dimensions of the field, the width of the path, and the area of the path. The left side of the equation represents the area of the path, which is calculated by subtracting the area of the field from the total area. The right side of the equation is the given area of the path, 1024 square meters. This equation is a quadratic equation in x, which means it involves a term with x squared. Solving this equation will give us the value(s) of x that satisfy the given conditions. Before we jump into solving, let's simplify the equation to make it easier to work with.

Solving the Quadratic Equation

The next step is to solve the quadratic equation we derived in the previous section: (70 + 2x)(50 + 2x) - 3500 = 1024. First, we need to expand the product on the left side of the equation. Expanding (70 + 2x)(50 + 2x) gives us 7050 + 702x + 502x* + 2x * 2x = 3500 + 140x + 100x + 4x^2 = 4x^2 + 240x + 3500. Substituting this back into the equation, we get 4x^2 + 240x + 3500 - 3500 = 1024. This simplifies to 4x^2 + 240x = 1024. By expanding and simplifying the equation, we've transformed it into a standard quadratic form, making it easier to solve.

Now, let's rearrange the equation to the standard quadratic form ax^2 + bx + c = 0. Subtracting 1024 from both sides of the equation gives us 4x^2 + 240x - 1024 = 0. To simplify further, we can divide the entire equation by 4, which gives us x^2 + 60x - 256 = 0. This simplified quadratic equation is now in a form that we can easily solve using various methods, such as factoring, completing the square, or using the quadratic formula. We will use factoring in this case. We look for two numbers that multiply to -256 and add to 60. These numbers are 64 and -4. Therefore, we can factor the quadratic equation as (x + 64)(x - 4) = 0. Setting each factor equal to zero gives us two possible solutions for x: x + 64 = 0 or x - 4 = 0. Solving these equations gives us x = -64 or x = 4. Since the width of the path cannot be negative, we discard the solution x = -64. Therefore, the width of the path is x = 4 meters. This step-by-step solution demonstrates the application of algebraic techniques to solve a geometric problem.

Verifying the Solution

It's always a good practice to verify the solution we've obtained to ensure its correctness. In this case, we found that the width of the path, x, is 4 meters. To verify this, we can substitute x = 4 back into the original equation or recalculate the area of the path using this width. If our solution is correct, the calculated area of the path should match the given area, which is 1024 square meters. This verification step helps to catch any potential errors in our calculations or reasoning.

Let's recalculate the area of the path using x = 4 meters. The outer dimensions of the rectangle formed by the field and the path are (70 + 24) meters and (50 + 24) meters, which are 78 meters and 58 meters, respectively. The total area enclosed by the outer rectangle is 78 * 58 = 4524 square meters. The area of the field is 70 * 50 = 3500 square meters. The area of the path is the difference between the total area and the area of the field, which is 4524 - 3500 = 1024 square meters. This matches the given area of the path, confirming that our solution of x = 4 meters is correct. This verification process reinforces the accuracy and reliability of our solution.

Conclusion

In conclusion, we have successfully determined the width of the concrete path surrounding the rectangular field. By carefully translating the problem statement into a mathematical equation, we were able to solve for the unknown width. We found that the width of the path is 4 meters. This problem demonstrates the power of combining geometric concepts with algebraic techniques to solve real-world problems. The process involved understanding the relationship between areas and dimensions, setting up a quadratic equation, solving the equation, and verifying the solution. This comprehensive approach is essential for problem-solving in mathematics and related fields.

This exercise highlights the importance of mathematical reasoning and problem-solving skills in various applications. Whether it's calculating areas for landscaping projects or determining dimensions for construction, the principles and techniques we've applied here are invaluable. The ability to translate real-world scenarios into mathematical models and solve them is a crucial skill in many disciplines. We hope this article has provided a clear and insightful explanation of the problem-solving process and has enhanced your understanding of geometry and algebra. Remember, practice and perseverance are key to mastering these skills.