Rectangle Dimensions Problem Solving Perimeter And Length Relationship

A rectangle has a perimeter of 806 feet. The length is 173 feet greater than the width. Determine the rectangle's width and length.

Embark on a mathematical journey to determine the precise width and length of a rectangular field, a classic problem that seamlessly blends geometry and algebra. This exploration will delve into the core concepts of perimeter, variable representation, equation formation, and problem-solving strategies, providing a comprehensive understanding of how mathematical principles can be applied to real-world scenarios.

Problem Statement

Imagine a rectangular field, a verdant expanse enclosed by boundaries that define its shape and size. We are presented with a crucial piece of information: the perimeter of this field measures 806 feet. This perimeter, the total distance around the rectangle, serves as the foundation for our mathematical quest. Adding a layer of complexity, we are informed that the length of the field surpasses its width by a significant 173 feet. This relationship between length and width introduces a variable element, challenging us to decipher the individual dimensions.

Our mission is clear: to unravel the mystery of the field's dimensions, to precisely calculate both its width and its length. This endeavor requires us to translate the given information into mathematical expressions, construct equations that capture the essence of the problem, and employ algebraic techniques to arrive at the sought-after solutions. Let us embark on this journey, armed with the tools of mathematics, to unlock the secrets of the rectangular field.

Defining Variables and Setting up Equations

To effectively tackle this problem, we must first translate the given information into the language of mathematics. This involves defining variables to represent the unknown quantities, the width and length of the rectangular field. Let us denote the width of the field as "w" feet. This simple assignment serves as the cornerstone of our algebraic representation.

Building upon this foundation, we can express the length of the field in terms of the width. We are informed that the length exceeds the width by 173 feet. Therefore, we can represent the length as "w + 173" feet. This expression captures the relationship between the two dimensions, paving the way for equation formation.

Now, let us turn our attention to the perimeter, a crucial piece of information that links the width and length. The perimeter of a rectangle is defined as the total distance around its boundary, which can be calculated by adding up the lengths of all four sides. In a rectangle, opposite sides are equal in length. Therefore, the perimeter can be expressed as 2 * (width) + 2 * (length).

We are given that the perimeter of the field is 806 feet. Substituting the expressions we have derived for width and length, we arrive at the following equation:

2 * (w) + 2 * (w + 173) = 806

This equation encapsulates the essence of the problem, linking the unknown width to the known perimeter through the relationship between length and width. With this equation in hand, we are poised to embark on the next phase of our mathematical journey: solving for the unknown width.

Solving the Equation

Our equation, 2 * (w) + 2 * (w + 173) = 806, stands as the key to unlocking the dimensions of the rectangular field. To isolate the unknown variable, "w", and determine its value, we must employ the principles of algebraic manipulation. Let us embark on this step-by-step process:

1. Distribute: Begin by distributing the 2 in the second term of the equation:

   2w + 2w + 346 = 806

2. Combine Like Terms: Combine the "w" terms on the left side of the equation:

   4w + 346 = 806

3. Isolate the Variable Term: Subtract 346 from both sides of the equation to isolate the term containing "w":

   4w = 806 - 346

   4w = 460

4. Solve for the Variable: Divide both sides of the equation by 4 to solve for "w":

   w = 460 / 4

   w = 115

With this final step, we have successfully determined the value of "w", the width of the rectangular field. The width measures 115 feet. Now, we can use this value to calculate the length of the field.

Calculating the Length

Having determined the width of the rectangular field to be 115 feet, we are now equipped to calculate its length. Recall that we established a relationship between the length and width earlier in our analysis: the length is 173 feet more than the width.

Using this relationship, we can express the length as:

Length = w + 173

Substituting the value of w (115 feet) into this equation, we obtain:

Length = 115 + 173

Length = 288

Therefore, the length of the rectangular field measures 288 feet. We have successfully determined both the width and the length, completing our mathematical exploration.

Presenting the Solution

We have embarked on a journey through the realms of geometry and algebra, and our efforts have yielded a definitive solution to the problem at hand. Let us now present our findings in a clear and concise manner:

The width of the rectangular field is 115 feet. The length of the rectangular field is 288 feet.

These values, meticulously derived through the application of mathematical principles, accurately represent the dimensions of the field. We have successfully translated a word problem into a mathematical framework, solved the resulting equations, and arrived at a concrete solution.

Conclusion

This exploration has showcased the power of mathematics in unraveling real-world problems. By carefully translating the given information into mathematical expressions, constructing equations, and employing algebraic techniques, we have successfully determined the dimensions of a rectangular field. This exercise serves as a testament to the versatility and applicability of mathematical principles in various domains.

From the initial problem statement to the final solution, we have traversed a path of logical reasoning and mathematical precision. The variables we defined, the equations we formulated, and the calculations we performed have all contributed to a comprehensive understanding of the problem and its solution. This journey underscores the importance of a systematic approach to problem-solving, where each step builds upon the previous one, leading to a conclusive and accurate answer.

The concepts explored in this problem, such as perimeter, variable representation, and equation solving, are fundamental building blocks in mathematics and have far-reaching applications in various fields, including engineering, physics, and computer science. By mastering these concepts, we equip ourselves with the tools to tackle a wide range of challenges, both within and beyond the realm of mathematics.

This exploration serves as a reminder that mathematics is not merely an abstract subject confined to textbooks and classrooms. It is a powerful tool that empowers us to understand and solve problems in the world around us. As we continue our mathematical journey, let us embrace the challenge of new problems, confident in our ability to apply the principles we have learned to unravel their mysteries.