In Exercises 36–43, use the five-step strategy for solving word problems. The length of a rectangular field is 6 yards less than triple the width. If the perimeter of the field is 340 yards, what are its dimensions?

Verified step by step guidance

1. Define the variables: Let the width of the rectangular field be denoted as \( w \) (in yards). Since the length is described as 6 yards less than triple the width, the length can be expressed as \( 3w - 6 \).
2. Recall the formula for the perimeter of a rectangle: \( P = 2 \times \text{length} + 2 \times \text{width} \). Substitute the given perimeter (340 yards) and the expressions for length and width into the formula: \( 340 = 2(3w - 6) + 2w \).
3. Simplify the equation: Distribute and combine like terms. Start by expanding \( 2(3w - 6) \) to get \( 6w - 12 \). Then, the equation becomes \( 340 = 6w - 12 + 2w \). Combine \( 6w \) and \( 2w \) to get \( 8w \), resulting in \( 340 = 8w - 12 \).
4. Solve for \( w \): Add 12 to both sides to isolate the term with \( w \), giving \( 352 = 8w \). Then, divide both sides by 8 to find \( w \).
5. Find the dimensions: Once \( w \) (the width) is determined, substitute it back into the expression for the length, \( 3w - 6 \), to calculate the length. The dimensions of the field are the width and the length.

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the rectangle, calculated by the formula P = 2L + 2W, where L is the length and W is the width. Understanding this formula is essential for solving problems related to the dimensions of rectangular shapes, as it allows you to relate the length and width to the given perimeter.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In this problem, the relationship between the length and width is expressed as L = 3W - 6. Recognizing how to form and manipulate these expressions is crucial for setting up equations that represent the problem's conditions.

Solving Linear Equations

Solving linear equations involves finding the value of variables that satisfy the equation. In this context, once the perimeter equation is set up using the expressions for length and width, you will need to solve for W and then use that value to find L. Mastery of this concept is vital for determining the dimensions of the field.