Questions: Translate the following phrase into an algebraic expression using the variable w. Do not simplify. the perimeter of a rectangle if the width is w centimeters and the length is 2 cm more than twice the width.

Solution Steps

To translate the given phrase into an algebraic expression, we need to identify the expressions for the width and length of the rectangle. The width is given as \( w \) cm, and the length is 2 cm more than twice the width, which can be expressed as \( 2w + 2 \) cm. The perimeter of a rectangle is calculated as \( 2 \times (\text{length} + \text{width}) \).

1. Define the width as \( w \).
2. Define the length as \( 2w + 2 \).
3. Use the perimeter formula for a rectangle: \( 2 \times (\text{length} + \text{width}) \).

Let the width of the rectangle be \( w \) cm. The length of the rectangle is given as 2 cm more than twice the width, which can be expressed as: \[ \text{Length} = 2w + 2 \]

The formula for the perimeter \( P \) of a rectangle is given by: \[ P = 2 \times (\text{Length} + \text{Width}) \] Substituting the expressions for length and width, we have: \[ P = 2 \times ((2w + 2) + w) \]

Now, we can simplify the expression for the perimeter: \[ P = 2 \times (2w + 2 + w) = 2 \times (3w + 2) = 6w + 4 \]

Final Answer