Question

a rectangular garden is designed to be 4 ft longer than it is wide. let x represent the width of the garden. write an expression for the perimeter in terms of the width x. write the expression in simplest form.
perimeter = 2x + 8

Explanation:

Step1: Determine length from width

Width is \( x \), length is \( x + 4 \) (since 4 ft longer than width).

Step2: Recall perimeter formula for rectangle

Perimeter \( P = 2\times(\text{length} + \text{width}) \). Substitute length and width: \( P = 2\times(x + (x + 4)) \).

Step3: Simplify the expression

First, simplify inside the parentheses: \( x + x + 4 = 2x + 4 \). Then multiply by 2: \( 2\times(2x + 4) = 4x + 8 \)? Wait, no, wait—wait, original thought might have error. Wait, let's redo: Wait, width is \( x \), length is \( x + 4 \). Perimeter of rectangle is \( 2(\text{length} + \text{width}) = 2(x + (x + 4)) = 2(2x + 4) = 4x + 8 \). But the given answer is \( 2x + 8 \), which is wrong. Wait, maybe misread. Wait, no—wait, maybe the problem was misstated? Wait, no, let's check again. Wait, if width is \( x \), length is \( x + 4 \). Perimeter is \( 2(x) + 2(x + 4) = 2x + 2x + 8 = 4x + 8 \). So the given answer \( 2x + 8 \) is incorrect. The correct expression should be \( 4x + 8 \). But the user provided a box with \( 2x + 8 \), but that's wrong. Wait, maybe I misread the problem. Wait, the problem says "4 ft longer than it is wide"—so length is \( x + 4 \). Perimeter is \( 2(\text{width} + \text{length}) = 2(x + x + 4) = 2(2x + 4) = 4x + 8 \). So the correct answer is \( 4x + 8 \), not \( 2x + 8 \).

Wait, but the user's image shows "Perimeter = 2x + 8" as the answer, but that's incorrect. Let's re-express the correct steps:

Step1: Define width and length

Let width \( = x \), then length \( = x + 4 \) (since length is 4 ft longer than width).

Step2: Apply perimeter formula for rectangle

Perimeter of a rectangle is \( P = 2\times(\text{width} + \text{length}) \). Substitute width \( x \) and length \( x + 4 \):
\( P = 2(x + (x + 4)) \)

Step3: Simplify the expression

First, combine like terms inside the parentheses: \( x + x + 4 = 2x + 4 \).
Then multiply by 2: \( 2\times(2x + 4) = 4x + 8 \).

Answer:

The correct expression for the perimeter is \( \boldsymbol{4x + 8} \) (the given \( 2x + 8 \) is incorrect).