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. Step 1: Identify what we are asked to find. In this case, we are asked to find the dimensions of the rectangular field, which are its length and width.
2. Step 2: Translate the word problem into mathematical equations. Let's denote the width of the field as 'w' (in yards). According to the problem, the length of the field is 6 yards less than triple the width, which can be written as '3w - 6'. The formula for the perimeter of a rectangle is 2*(length + width). So, we can write the equation as 2*(w + 3w - 6) = 340.
3. Step 3: Simplify the equation. Combine like terms and distribute the 2 across the terms inside the parentheses to get 8w - 12 = 340.
4. Step 4: Solve the equation for 'w'. Add 12 to both sides of the equation to isolate the term with 'w' on one side, then divide by 8 to find the value of 'w'.
5. Step 5: Substitute the value of 'w' back into the equation for the length (3w - 6) to find the length of the field.

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 of the field is expressed algebraically, where the length is defined as 6 yards less than triple the width. This understanding is crucial for setting up equations that represent the problem.

Solving Linear Equations

Solving linear equations involves finding the value of variables that satisfy the equation. In this context, once the relationships between length and width are established, you will set up and solve a linear equation based on the perimeter. Mastery of this concept is vital for determining the dimensions of the rectangular field.