Question

1. you expand a rectangular garden so the perimeter is now twice the perimeter of the old garden. the expression 12w + 16 represents the perimeter of the new garden, where w represents the width of the old garden.

a. write an expression that represents the perimeter of the old garden. justify your answer.

b. write an expression that represents the area of the old garden.

Explanation:

Part a: Perimeter of Old Garden

Step1: Recall Perimeter Formula

The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \), where \( l \) is length and \( w \) is width.

Step2: Analyze Given Perimeter

We know the new perimeter is \( 12w + 16 \), and it's twice the old perimeter. Let \( P_{\text{old}} \) be the old perimeter. So \( 2P_{\text{old}} = 12w + 16 \).

Step3: Solve for \( P_{\text{old}} \)

Divide both sides by 2: \( P_{\text{old}}=\frac{12w + 16}{2}=6w + 8 \).

Step1: Recall Perimeter - Length Relationship

From \( P_{\text{old}} = 2(l + w)=6w + 8 \), divide by 2: \( l + w = 3w + 4 \). Then solve for \( l \): \( l=3w + 4 - w = 2w + 4 \).

Step2: Recall Area Formula

The area \( A \) of a rectangle is \( A = l\times w \). Substitute \( l = 2w + 4 \) and \( w \): \( A=(2w + 4)w = 2w^{2}+4w \).

Answer:

The expression for the perimeter of the old garden is \( 6w + 8 \).

Part b: Area of Old Garden