Solving Rectangle Dimensions Finding The Width When Length And Perimeter Are Known

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Finding the dimensions of geometric shapes is a common task in mathematics and real-world applications. In this article, we will delve into a specific problem involving a rectangle, where the length is defined in terms of its width, and the perimeter is known. We will use an algebraic equation to solve for the width, providing a clear, step-by-step explanation to enhance understanding.

Understanding the Problem

Before diving into the solution, let's dissect the problem statement. We are given that the length of the rectangle is six feet more than two times the width. This can be mathematically expressed as L = 2_W_ + 6, where L represents the length and W represents the width. We are also given that the perimeter of the rectangle is 72 feet. The perimeter of a rectangle is the total distance around its sides, which can be calculated using the formula P = 2_L_ + 2_W_, where P represents the perimeter.

The problem provides us with an equation, 2_W_ + 2(2_W_ + 6) = 72, which we will use to solve for the width, W. This equation combines the information about the perimeter and the relationship between the length and width.

Setting up the Equation

The equation 2_W_ + 2(2_W_ + 6) = 72 is derived from the perimeter formula and the relationship between the length and width. Let's break down how this equation is formed:

  1. Perimeter Formula: We know that the perimeter P of a rectangle is given by P = 2_L_ + 2_W_.
  2. Length in terms of Width: We are given that the length L is six feet more than two times the width, which can be written as L = 2_W_ + 6.
  3. Substitution: We substitute the expression for L from step 2 into the perimeter formula from step 1: P = 2(2_W_ + 6) + 2_W_.
  4. Given Perimeter: We are given that the perimeter P is 72 feet. So, we substitute 72 for P in the equation: 72 = 2(2_W_ + 6) + 2_W_.
  5. Rearranging the Equation: Finally, we can rearrange the equation to match the one given in the problem: 2_W_ + 2(2_W_ + 6) = 72.

This equation now represents the problem mathematically and sets the stage for solving for the width.

Solving the Equation Step-by-Step

Now, let's walk through the process of solving the equation 2_W_ + 2(2_W_ + 6) = 72 step-by-step:

  1. Distribute: Start by distributing the 2 in the second term: 2_W_ + 2(2_W_ + 6) = 2_W_ + 4_W_ + 12.
  2. Rewrite the Equation: Now the equation looks like this: 2_W_ + 4_W_ + 12 = 72.
  3. Combine Like Terms: Combine the terms with W: 6_W_ + 12 = 72.
  4. Isolate the Variable Term: Subtract 12 from both sides of the equation to isolate the term with W: 6_W_ + 12 - 12 = 72 - 12, which simplifies to 6_W_ = 60.
  5. Solve for W: Divide both sides by 6 to solve for W: 6_W_ / 6 = 60 / 6, which gives us W = 10.

Therefore, the width of the rectangle is 10 feet.

Verifying the Solution

It's always a good practice to verify the solution to ensure accuracy. To do this, we can plug the value of W back into the original equation and the formulas for length and perimeter.

  1. Calculate the Length: Using the formula L = 2_W_ + 6, substitute W = 10: L = 2(10) + 6 = 20 + 6 = 26 feet.
  2. Calculate the Perimeter: Using the perimeter formula P = 2_L_ + 2_W_, substitute L = 26 and W = 10: P = 2(26) + 2(10) = 52 + 20 = 72 feet.

Since the calculated perimeter matches the given perimeter of 72 feet, our solution for the width, W = 10 feet, is correct.

Importance of Understanding Rectangle Dimensions

Understanding how to calculate the dimensions of rectangles is crucial in various fields, ranging from basic geometry to real-world applications. Whether you're determining the amount of fencing needed for a rectangular garden, calculating the area of a room for flooring, or designing a rectangular structure, the principles of perimeter, area, and the relationship between length and width are fundamental.

In architecture and construction, accurate dimension calculations are essential for designing buildings, rooms, and other structures. Knowing the perimeter helps in determining the amount of materials needed for walls, flooring, and roofing. Similarly, the area calculation is crucial for space planning and material estimation.

In interior design, understanding dimensions is vital for space utilization and furniture placement. Interior designers use these concepts to create functional and aesthetically pleasing spaces.

In everyday life, calculating dimensions can help with various tasks, such as determining the size of a picture frame, estimating the amount of paint needed for a wall, or even arranging furniture in a room.

Common Mistakes to Avoid

While solving problems involving rectangle dimensions, it's important to avoid common mistakes that can lead to incorrect answers. Here are some pitfalls to watch out for:

  1. Misunderstanding the Formulas: Ensure you correctly understand and apply the formulas for perimeter and area. Confusing the formulas or using them incorrectly is a common mistake.
  2. Incorrectly Setting up the Equation: When translating word problems into algebraic equations, pay close attention to the relationships between the variables. Misinterpreting the problem statement can lead to an incorrect equation.
  3. Arithmetic Errors: Simple arithmetic errors during calculations can result in wrong answers. Double-check your calculations to minimize mistakes.
  4. Forgetting Units: Always include the appropriate units in your answers (e.g., feet, meters, inches). Forgetting units can lead to misunderstandings and practical errors.
  5. Not Verifying the Solution: As demonstrated earlier, verifying your solution is crucial. Plugging the solution back into the original equation and formulas can help catch errors.

Conclusion

In this article, we tackled a problem involving the dimensions of a rectangle, where the length was expressed in terms of the width, and the perimeter was known. We successfully solved for the width using an algebraic equation and verified our solution. Understanding the concepts of perimeter, area, and the relationships between length and width is essential in various mathematical and real-world contexts.

By following a systematic approach, setting up equations correctly, and avoiding common mistakes, you can confidently solve problems involving rectangle dimensions and apply these skills in practical situations.