Exercises 141–143 will help you prepare for the material covered in the next section. If the width of a rectangle is represented by x and the length is represented by x + 200, write a simplified algebraic expression that models the rectangle's perimeter.

Verified step by step guidance

1. insert step 1: Recall the formula for the perimeter of a rectangle, which is given by P = 2 \times (\text{length} + \text{width}).
2. insert step 2: Substitute the given expressions for the length and width into the formula. Here, the length is x + 200 and the width is x.
3. insert step 3: Write the expression for the perimeter: P = 2 \times ((x + 200) + x).
4. insert step 4: Simplify the expression inside the parentheses: (x + 200) + x = 2x + 200.
5. insert step 5: Distribute the 2 to both terms inside the parentheses to get the simplified expression for the perimeter: P = 2 \times (2x + 200).

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the rectangle, calculated by adding together the lengths of all four sides. The formula for the perimeter (P) is P = 2(length + width). Understanding this formula is essential for modeling the perimeter based on given dimensions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In this context, the expression will involve the variable x, representing the width, and will be used to express the perimeter in terms of x. Simplifying these expressions is a key skill in algebra.

Substitution

Substitution is the process of replacing a variable in an expression with a specific value or another expression. In this problem, we substitute the length of the rectangle, which is expressed as (x + 200), into the perimeter formula. This allows us to create a single expression that represents the perimeter in terms of x.