Finding the Dimensions of a Rectangle Given Its Perimeter and Length-Width Relationship

In this article, we will go through the process of finding the dimensions of a rectangle when we are given its perimeter and a relationship between its length and width. This problem involves algebraic manipulation and the application of geometric principles. Let's walk through the steps to solve for the width and length of the rectangle.

Problem Statement

We are given a rectangle where the length is 7 inches more than its width. If the perimeter of the rectangle is 66 inches, what are the dimensions of the rectangle?

Step-by-Step Solution

Step 1: Define Variables

Let the width of the rectangle be w inches. The length of the rectangle can be expressed as l w 7 inches.

Step 2: Perimeter Formula

The formula for the perimeter of a rectangle is given by:

[ P 2l 2w ]

Step 3: Substitute Length Expression

Substitute the expression for l into the perimeter formula:

[ 66 2(textbf{w} 7) 2w ]

This simplifies to:

[ 66 2textbf{w} 14 2w ]

Step 4: Combine Like Terms

Combine the like terms (the terms involving w):

[ 66 4w 14 ]

Step 5: Isolate the Variable

Subtract 14 from both sides to isolate the variable term:

[ 52 4w ]

Divide both sides by 4 to solve for w:

[ w 13 ]

Step 6: Find the Length

Now that we have the width, we can find the length:

[ l w 7 13 7 20 ]

Thus, the dimensions of the rectangle are:

- Width: 13 inches - Length: 20 inches

Double Check

To ensure the calculation is correct, we can double-check the perimeter:

[ 2(13) 2(20) 26 40 66 ] inches, which confirms that our solution is accurate.

Conclusion

The process of solving for the dimensions of a rectangle given its perimeter and a relationship between its length and width involves setting up an equation based on the perimeter formula and then solving for the unknown variables. By following these steps, we can find the exact dimensions of the rectangle.

Key Points:

- The relationship between length and width gives us an equation to work with. - The perimeter formula helps us form an equation that can be solved algebraically. - Double-checking the solution ensures accuracy.

Understanding these concepts is crucial for solving similar geometric problems. Whether you are designing a layout, planning a space, or simply solving a math problem, knowing how to apply these principles can be incredibly useful.

Keywords

Rectangle dimensions, Perimeter calculation, Length-width relationship