Solve Perimeter Equation 2l + 2w = 62 Find Rectangle Width

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In this mathematical exploration, we delve into a classic problem involving the perimeter of a rectangle. The scenario presents us with a rectangle where the length (l) is given as 20 feet, and the perimeter is defined by the equation 2l + 2w = 62, where w represents the width. Our mission is to dissect this equation, solve for the unknown width (w), and then rigorously evaluate the given statements to determine their veracity. This problem is not just about crunching numbers; it's about understanding the relationships between geometric properties, algebraic equations, and real-world constraints. By meticulously analyzing each step, we'll uncover the correct solution and gain a deeper appreciation for the power of mathematical modeling.

Setting the Stage: Understanding the Problem

Before we jump into the calculations, let's take a moment to fully grasp the problem at hand. We have a rectangle, a fundamental geometric shape characterized by its two pairs of equal sides and four right angles. The perimeter of any shape is the total distance around its outer boundary. In the case of a rectangle, the perimeter is calculated by adding up the lengths of all four sides. Since a rectangle has two lengths and two widths, the formula for its perimeter is 2l + 2w, where l is the length and w is the width. This is a crucial concept to internalize, as it forms the bedrock of our solution. Understanding the relationship between the sides of a rectangle and its perimeter allows us to translate real-world scenarios into mathematical equations, which we can then solve to gain insights.

The problem gives us two key pieces of information: the length (l) of the rectangle is 20 feet, and the perimeter is 62 feet. This information is elegantly summarized in the equation 2l + 2w = 62. This equation is the bridge between the geometric description of the rectangle and the algebraic tools we'll use to find the width. By substituting the known length into the equation, we can isolate the unknown width and solve for its value. This process of translating a real-world problem into a mathematical equation is a cornerstone of mathematical modeling, a powerful technique used across countless disciplines.

The Solution Unveiled: Finding the Value of w

Now, let's roll up our sleeves and tackle the equation head-on. Our goal is to isolate w on one side of the equation, thereby revealing its value. The equation we're working with is 2l + 2w = 62. The first step in this algebraic dance is to substitute the given value of the length, l = 20 feet, into the equation. This gives us 2(20) + 2w = 62. This substitution is a critical step, as it allows us to reduce the equation from two unknowns (l and w) to just one (w), making it solvable.

Next, we perform the multiplication: 2(20) = 40. Our equation now looks like this: 40 + 2w = 62. To further isolate w, we need to get rid of the 40 on the left side of the equation. We achieve this by subtracting 40 from both sides of the equation. This is a fundamental principle of algebraic manipulation: whatever operation you perform on one side of the equation, you must perform on the other side to maintain the balance. Subtracting 40 from both sides gives us 2w = 62 - 40, which simplifies to 2w = 22.

We're almost there! The final step in our quest to find w is to divide both sides of the equation by 2. This isolates w completely, revealing its value. Dividing both sides of 2w = 22 by 2 gives us w = 22 / 2, which simplifies to w = 11. Thus, we have successfully solved for the width of the rectangle, finding that w is equal to 11 feet. This is the numerical answer to our problem, but our journey doesn't end here. We must now analyze the given statements to see which ones hold true in light of our solution.

With the value of w firmly in our grasp, we now turn our attention to the statements provided in the problem. Each statement makes a claim about the solution, and our task is to determine whether these claims align with our calculated value of w = 11 feet. This is a crucial step in the problem-solving process, as it ensures that our answer not only satisfies the equation but also makes sense in the context of the original problem.

Statement A: The value of w is 10 feet.

This statement is a direct claim about the value of w. To evaluate its truthfulness, we simply compare it to our calculated value. We found that w = 11 feet, while the statement claims w = 10 feet. These values are clearly different. Therefore, Statement A is false. It's important to note that even a small discrepancy can render a statement false in mathematics. Precision is paramount, and our calculations have shown that the width is definitively 11 feet, not 10 feet.

Statement B: The value of w can be zero.

This statement delves into the realm of possibilities, asking whether w could be zero. While mathematically, substituting w = 0 into the equation 2l + 2w = 62 might seem to yield a valid result (2(20) + 2(0) = 40, which is not 62, indicating an issue), we must consider the real-world context of the problem. We are dealing with a rectangle, a geometric shape with specific properties. By definition, a rectangle must have a width greater than zero. A width of zero would imply that the rectangle has collapsed into a line segment, ceasing to be a two-dimensional shape. This is a crucial distinction – mathematical equations can sometimes produce solutions that are not physically or geometrically meaningful. In this case, w cannot be zero because it violates the fundamental definition of a rectangle. Therefore, Statement B is false.

Statement C: The value of w cannot be a negative number.

This statement probes the boundaries of possible values for w, specifically focusing on negative numbers. Similar to the previous statement, we must consider both the mathematical equation and the geometric context. While negative numbers can exist in mathematical solutions, they often lack physical meaning in geometric problems. A negative width for a rectangle is conceptually nonsensical. Width, as a measure of distance, cannot be negative. Just as you cannot have a negative length or a negative height, you cannot have a negative width. This is a fundamental principle of geometric measurement. Therefore, Statement C is true. The width of a rectangle, in the real world, must be a positive value.

In this comprehensive exploration, we've not only solved for the width of the rectangle but also critically evaluated each statement provided. Our journey began with a clear understanding of the perimeter equation, 2l + 2w = 62, and the given length of 20 feet. Through careful algebraic manipulation, we determined that the width, w, is 11 feet. This numerical solution served as the foundation for our analysis of the statements.

Our meticulous evaluation revealed that Statement A, claiming the width is 10 feet, is false. Statement B, suggesting the width could be zero, is also false, as it violates the geometric definition of a rectangle. Finally, Statement C, asserting that the width cannot be a negative number, is true, aligning with the fundamental principles of geometric measurement.

This problem highlights the importance of not just solving equations but also interpreting the solutions within the context of the original problem. Mathematical solutions must be both mathematically valid and physically meaningful. By considering the geometric constraints of the rectangle, we were able to confidently determine the truthfulness of each statement. This approach of combining algebraic techniques with geometric reasoning is a powerful tool for problem-solving in mathematics and beyond.

  • Perimeter Equation
  • Solving for Width
  • Rectangle Dimensions
  • Algebraic Manipulation
  • Geometric Constraints
  • Mathematical Modeling
  • Evaluating Statements
  • Truthfulness Analysis
  • Problem-Solving Process
  • Negative Number
  • Zero Value
  • Positive Value
  • Comprehensive Exploration

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Solve Perimeter Equation 2l + 2w = 62 Find Rectangle Width