Finding The Length Of A Rectangle Step-by-Step Guide

In the realm of mathematics, particularly geometry, rectangles hold a fundamental position. These four-sided figures, with their right angles and parallel sides, appear in countless applications, from architectural designs to everyday objects. Understanding the properties of rectangles, such as their area and perimeter, is crucial for solving a variety of problems. In this comprehensive guide, we will delve into a specific problem involving a rectangle, where the length is related to the width, and the area is known. Our goal is to determine the length of the rectangle, and we will explore the steps involved in setting up the equation, solving for the unknown dimensions, and arriving at the final answer.

Understanding the Problem

To begin, let's carefully analyze the problem statement. We are given that the length of a rectangle is 4 yards more than its width. This establishes a relationship between the two dimensions of the rectangle. We are also given that the area of the rectangle is 60 square yards. This provides us with a numerical value that connects the length and width through the area formula. Our objective is to find the length of the rectangle. This means we need to determine the specific value of the longer side of the rectangle.

To tackle this problem effectively, we will employ algebraic techniques. This involves representing the unknown quantities (length and width) with variables, setting up an equation based on the given information, and solving the equation to find the values of the variables. Once we have the values of the length and width, we can identify the length as the solution to our problem.

Setting up the Equation

In this section, we will translate the word problem into a mathematical equation. This is a crucial step in solving any mathematical problem, as it allows us to manipulate the information using algebraic rules and arrive at a solution. Let's start by defining variables to represent the unknown quantities.

Let w represent the width of the rectangle in yards. Since the length is 4 yards more than the width, we can represent the length as w + 4 yards. Now, we need to use the information about the area of the rectangle to form an equation. The area of a rectangle is given by the formula:

Area = Length × Width

We are given that the area is 60 square yards. Substituting the expressions for length and width into the area formula, we get:

60 = (w + 4) × w

This equation represents the situation described in the problem. It relates the width (w) of the rectangle to its area (60 square yards) through the given relationship between length and width. Now, we need to solve this equation for w.

Solving the Equation

To solve the equation 60 = (w + 4) × w, we first need to simplify it by expanding the right side:

60 = w² + 4w

Now, we have a quadratic equation. To solve a quadratic equation, we generally want to set it equal to zero. So, we subtract 60 from both sides of the equation:

0 = w² + 4w - 60

This is a standard quadratic equation in the form of ax² + bx + c = 0, where a = 1, b = 4, and c = -60. There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a suitable method.

We need to find two numbers that multiply to -60 and add up to 4. These numbers are 10 and -6. So, we can factor the quadratic equation as follows:

0 = (w + 10)(w - 6)

Now, we can use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible solutions for w:

w + 10 = 0 or w - 6 = 0

Solving these equations, we get:

w = -10 or w = 6

Since the width of a rectangle cannot be negative, we discard the solution w = -10. Therefore, the width of the rectangle is w = 6 yards.

Finding the Length

Now that we have found the width of the rectangle, we can determine the length using the relationship we established earlier: length = w + 4. Substituting the value of w = 6, we get:

Length = 6 + 4 = 10 yards

So, the length of the rectangle is 10 yards. This is the final answer to our problem.

Verifying the Solution

It's always a good practice to verify our solution to ensure that it satisfies the given conditions. We found that the width of the rectangle is 6 yards and the length is 10 yards. Let's check if these values are consistent with the information provided in the problem.

First, we need to verify that the length is 4 yards more than the width. Indeed, 10 yards is 4 yards more than 6 yards. Second, we need to verify that the area of the rectangle is 60 square yards. The area is given by:

Area = Length × Width = 10 yards × 6 yards = 60 square yards

This confirms that our solution is correct. The length of the rectangle is 10 yards, and the width is 6 yards, satisfying both the relationship between length and width and the given area.

Generalizing the Approach

The approach we used to solve this problem can be generalized to solve other similar problems involving rectangles and their dimensions. The key steps are:

1. Define variables: Assign variables to represent the unknown quantities, such as length and width.
2. Establish relationships: Identify the relationships between the variables based on the information given in the problem. For example, the length might be a certain amount more or less than the width.
3. Set up an equation: Use the given information and the area formula (Area = Length × Width) to form an equation.
4. Solve the equation: Solve the equation for the unknown variables. This might involve factoring, using the quadratic formula, or other algebraic techniques.
5. Interpret the solution: Choose the appropriate solution based on the context of the problem (e.g., discard negative solutions for length or width).
6. Verify the solution: Check if the solution satisfies the given conditions in the problem.

By following these steps, you can confidently solve a wide range of problems involving rectangles and their dimensions.

Real-World Applications

The concepts and techniques we have discussed in this guide have numerous real-world applications. Rectangles are fundamental shapes in architecture, engineering, and design. Understanding how to calculate their area and dimensions is essential for tasks such as:

• Designing buildings and structures: Architects and engineers need to calculate the area of rooms, walls, and other rectangular spaces to determine material requirements and ensure structural integrity.
• Laying out gardens and landscapes: Landscapers use rectangles to design flower beds, lawns, and other outdoor spaces, calculating the area to determine the amount of soil, mulch, or other materials needed.
• Manufacturing and packaging: Manufacturers use rectangles to design boxes, containers, and other packaging materials, calculating the area to minimize material waste and ensure efficient storage and transportation.
• Computer graphics and image processing: Rectangles are used to represent windows, images, and other graphical elements in computer applications, and their dimensions and positions need to be calculated accurately.

These are just a few examples of the many real-world applications of the concepts we have discussed. By mastering these concepts, you will be well-equipped to tackle a variety of practical problems.

Conclusion

In this comprehensive guide, we have explored a problem involving a rectangle, where the length is related to the width, and the area is known. We have demonstrated how to set up an equation, solve for the unknown dimensions, and arrive at the final answer. We have also discussed the general approach to solving similar problems and highlighted the real-world applications of these concepts.

By understanding the properties of rectangles and mastering the techniques for solving related problems, you will enhance your mathematical skills and be better prepared to tackle a wide range of practical challenges. Remember to carefully analyze the problem statement, define variables, establish relationships, set up an equation, solve the equation, interpret the solution, and verify the solution. With practice and perseverance, you will become proficient in solving rectangle-related problems and applying these skills in various real-world contexts. The length of a rectangle is not just a mathematical concept; it is a fundamental element in many aspects of our lives.