Rectangle Width Calculation Given Length $\sqrt{50}$ And Perimeter $10\sqrt{2}+4\sqrt{10}$

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Introduction

In the realm of geometry, rectangles hold a fundamental place, characterized by their pairs of equal-length sides and four right angles. Understanding the dimensions of a rectangle—its length and width—is crucial for various applications, from calculating area and perimeter to solving complex geometric problems. This article delves into a specific problem involving a rectangle with a given length and perimeter, challenging us to determine its width. This problem not only tests our understanding of geometric formulas but also our ability to manipulate radicals and algebraic expressions. To truly grasp the solution, we will first lay the groundwork by defining the key concepts and formulas related to rectangles. Understanding these basics is not just about solving a problem; it's about building a solid foundation in geometry that will aid in tackling more complex challenges in the future.

Defining Rectangles and Their Properties

A rectangle, at its core, is a quadrilateral—a four-sided polygon—where all four angles are right angles (90 degrees). This defining characteristic ensures that opposite sides of a rectangle are parallel and equal in length. The longer side is typically referred to as the length, while the shorter side is the width. These two dimensions are the building blocks for calculating two key properties of a rectangle: the perimeter and the area. The perimeter, as we'll explore further, is the total distance around the rectangle, while the area represents the space enclosed within its sides. Understanding these properties and how they relate to the length and width is essential for solving various geometric problems.

Perimeter: The Distance Around

The perimeter of any shape is the total length of its boundary. For a rectangle, this means adding up the lengths of all four sides. Since a rectangle has two sides of equal length (the lengths) and two other sides of equal length (the widths), the formula for the perimeter ( extit{P}) can be expressed as:

\textit{P} = 2 \times \text{length} + 2 \times \text{width}

This formula is a cornerstone in solving problems involving rectangles, especially when given the perimeter and one of the dimensions, as in our case. The perimeter essentially traces the outline of the rectangle, giving us a measure of the distance one would travel if they walked around its edges. Understanding this concept is not just about memorizing a formula; it's about visualizing the physical distance and how it relates to the dimensions of the rectangle.

Radicals in Geometry: A Necessary Tool

In many geometric problems, especially those involving lengths and distances, radicals (like square roots) often appear. This is because the Pythagorean theorem, a fundamental concept in geometry, frequently leads to radical expressions when calculating side lengths in right-angled triangles and other shapes. In our problem, the length and perimeter are given in terms of radicals, specifically square roots. Therefore, manipulating and simplifying radicals is a crucial skill for solving this problem. This involves understanding how to add, subtract, multiply, and divide radicals, as well as how to simplify them by factoring out perfect squares. The ability to work with radicals is not just a mathematical technique; it's a way to express and understand geometric relationships in a precise and meaningful way.

Problem Statement: Unveiling the Unknown Width

Now, let's formally state the problem we aim to solve. We are given a rectangle with a length of $\sqrt{50}$ units. The perimeter of this rectangle is specified as $10\sqrt{2} + 4\sqrt{10}$ units. Our primary objective is to determine the width of this rectangle. This problem challenges us to apply the perimeter formula in reverse, using the given perimeter and length to find the missing width. This requires not only an understanding of the formula but also algebraic manipulation skills to isolate the unknown variable. Furthermore, the presence of radicals adds another layer of complexity, necessitating the simplification of radical expressions to arrive at the final answer. Successfully solving this problem showcases a comprehensive understanding of both geometric principles and algebraic techniques.

Breaking Down the Given Information

To tackle this problem effectively, we must first dissect the information provided. We know the length ( extit{l}) of the rectangle is $\sqrt{50}$. This radical can be simplified, which we will do shortly. We also know the perimeter ( extit{P}) is $10\sqrt{2} + 4\sqrt{10}$. The key here is to recognize that the perimeter is the sum of all the sides, and since a rectangle has two pairs of equal sides, we can use this information along with the given length to find the width. This step is crucial because it sets the stage for applying the perimeter formula and solving for the unknown width. By carefully examining the given information, we can identify the necessary steps and strategies to solve the problem.

The Unknown: Defining the Width

Our ultimate goal is to find the width of the rectangle. Let's denote the width as extit{w}. This simple act of assigning a variable to the unknown is a fundamental step in algebra. It allows us to translate the geometric problem into an algebraic equation, which we can then solve using established techniques. The width, in this context, represents the shorter side of the rectangle and is the dimension we are trying to uncover. By clearly defining the unknown, we can focus our efforts on finding its value using the given information and the relevant formulas. This step is not just about assigning a letter; it's about framing the problem in a way that makes it solvable.

Solution: Finding the Missing Dimension

To determine the width of the rectangle, we will use the perimeter formula and the given information. This involves a series of steps, including simplifying radicals, substituting values into the formula, and solving the resulting equation. Each step is crucial in arriving at the correct answer, and understanding the logic behind each step is as important as the final solution itself. This process not only provides the answer but also reinforces our understanding of the relationship between perimeter, length, and width in a rectangle.

Step 1: Simplifying the Radical Length

The first step in solving this problem is to simplify the given length, which is $\sqrt{50}$. Simplifying radicals involves factoring out perfect squares from under the square root. In this case, 50 can be factored as $25 \times 2$, where 25 is a perfect square. Therefore, $\sqrt{50}$ can be rewritten as $\sqrt{25 \times 2}$. Using the property of radicals that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we get $\sqrt{25} \times \sqrt{2}$. Since $\sqrt{25} = 5$, the simplified length is $5\sqrt{2}$. This simplification makes the subsequent calculations easier and more manageable. Simplifying radicals is a fundamental skill in algebra and is often necessary to express answers in their simplest form.

Step 2: Applying the Perimeter Formula

Now that we have the simplified length, we can apply the perimeter formula. Recall that the perimeter of a rectangle is given by

\textit{P} = 2l + 2w

where extit{P} is the perimeter, extit{l} is the length, and extit{w} is the width. We are given that the perimeter extit{P} is $10\sqrt{2} + 4\sqrt{10}$, and we have found that the length extit{l} is $5\sqrt{2}$. Substituting these values into the formula, we get:

$10\sqrt{2} + 4\sqrt{10} = 2(5\sqrt{2}) + 2w$

This equation now relates the known perimeter and length to the unknown width. The next step is to simplify and solve this equation for extit{w}. This process involves algebraic manipulation and careful attention to the order of operations.

Step 3: Solving for the Width

To solve for the width, we first simplify the equation we obtained in the previous step:

$10\sqrt{2} + 4\sqrt{10} = 2(5\sqrt{2}) + 2w$

Multiplying out the term on the right side gives:

$10\sqrt{2} + 4\sqrt{10} = 10\sqrt{2} + 2w$

Now, we want to isolate the term with extit{w}. We can subtract $10\sqrt{2}$ from both sides of the equation:

$10\sqrt{2} + 4\sqrt{10} - 10\sqrt{2} = 10\sqrt{2} + 2w - 10\sqrt{2}$

This simplifies to:

$4\sqrt{10} = 2w$

Finally, to solve for extit{w}, we divide both sides of the equation by 2:

$\frac{4\sqrt{10}}{2} = \frac{2w}{2}$

Which gives us:

$2\sqrt{10} = w$

Therefore, the width of the rectangle is $2\sqrt{10}$ units. This result is the solution to our problem, and it represents the missing dimension of the rectangle.

Final Answer and Verification

Thus, the width of the rectangle is $2\sqrt{10}$. This corresponds to option (c) among the given choices. But before we definitively conclude, it’s crucial to verify our answer. Verification is a vital step in problem-solving, ensuring that our solution is not only mathematically correct but also logically consistent with the given information. To verify our answer, we will substitute the calculated width back into the perimeter formula and check if it matches the given perimeter.

Verification Process: Ensuring Accuracy

To verify our solution, we substitute the calculated width, $2\sqrt{10}$, back into the perimeter formula:

\textit{P} = 2l + 2w

We know that the length extit{l} is $5\sqrt{2}$, and we have found that the width extit{w} is $2\sqrt{10}$. Substituting these values, we get:

\textit{P} = 2(5\sqrt{2}) + 2(2\sqrt{10})

Simplifying this expression:

\textit{P} = 10\sqrt{2} + 4\sqrt{10}

This result matches the given perimeter in the problem statement. This confirms that our calculated width is correct. The verification process not only gives us confidence in our answer but also reinforces our understanding of the relationships between the dimensions and perimeter of a rectangle.

Conclusion

In this exploration, we successfully determined the width of a rectangle given its length and perimeter. The problem required us to apply the perimeter formula, simplify radicals, and solve algebraic equations. The process involved a step-by-step approach, starting with simplifying the given length, substituting values into the perimeter formula, solving for the unknown width, and finally, verifying the solution. The final answer, $2\sqrt{10}$, aligns with option (c) and was confirmed through the verification process. This exercise underscores the importance of a solid understanding of geometric principles and algebraic techniques in problem-solving. The ability to manipulate radicals, apply formulas, and solve equations are essential skills in mathematics and are applicable in various real-world scenarios. Moreover, the emphasis on verification highlights the significance of ensuring the accuracy and consistency of our solutions. By mastering these concepts and techniques, we can confidently tackle a wide range of geometric challenges and enhance our problem-solving abilities.