Solving for the Dimensions of a Rectangle Given Its Perimeter

Understanding how to calculate the dimensions of a rectangle based on its perimeter is a fundamental concept in mathematics, particularly in pre-algebra. This article will guide you through the process of finding the length and width of a rectangle given that its perimeter is 36 feet, and the length is 2 less than 3 times the width. We will explore multiple methods, including substitution and algebraic equations, to arrive at the correct answer.

Problem Statement

The problem at hand is to find the dimensions of a rectangle given that its perimeter is 36 feet. Additionally, we know that the length is 2 feet less than 3 times the width. Let's denote the width by ( W ) and the length by ( L ).

Solution Using Substitution Method

Step 1: Set Up the Equations

The perimeter of a rectangle is given by the formula: [ P 2L 2W ] Given that the perimeter ( P 36 ) feet, we can write: [ 2L 2W 36 ] The length ( L ) is 2 feet less than 3 times the width ( W ), so we have: [ L 3W - 2 ]

Step 2: Substituting the Length into the Perimeter Formula

Substitute ( L 3W - 2 ) into the perimeter formula:

[ 2(3W - 2) 2W 36 ]

Step 3: Simplify and Solve the Equation

Simplify the equation step by step:

Expand the equation: [ 6W - 4 2W 36 ] Combine like terms: [ 8W - 4 36 ] Add 4 to both sides: [ 8W 40 ] Solve for ( W ): [ W frac{40}{8} 5 ] feet

Step 4: Find the Length

Now that we have the width ( W 5 ) feet, we can find the length ( L ) using the relationship ( L 3W - 2 ):

[ L 3(5) - 2 15 - 2 13 ] feet

So, the length is 13 feet and the width is 5 feet.

Additional Methods

Algebraic Equation Method

We can also use the algebraic equation method to solve the problem. Let's denote the width as ( W ) and the length as ( L ) again:

The perimeter equation remains: [ 2L 2W 36 ] The relationship between the length and width is: [ L 3W - 2 ]

Substituting the length into the perimeter equation gives:

[ 2(3W - 2) 2W 36 ]

Simplifying this equation further:

Expand the equation: [ 6W - 4 2W 36 ] Combine like terms: [ 8W - 4 36 ] Add 4 to both sides: [ 8W 40 ] Solve for ( W ): [ W 5 ] feet Find the length: [ L 3(5) - 2 13 ] feet

Verification

To ensure the solution is correct, we can verify by checking the perimeter:

[ P 2L 2W 2(13) 2(5) 26 10 36 ] feet

This confirms our solution is accurate.

Conclusion

In this article, we have solved the problem of finding the dimensions of a rectangle with a perimeter of 36 feet and a length that is 2 feet less than 3 times the width using both the substitution method and algebraic equations. The final answer is that the length is 13 feet and the width is 5 feet.