Question

1. the diagram represents a flower border that is 3 feet wide surrounding a rectangular sitting area. write an expression in factored form that represents the area of the flower border.

Explanation:

Step1: Find the area of the outer rectangle

Let the length of the inner rectangular sitting area be \(l\) and the width be \(w\). The width of the flower - border is \(3\) feet. So the length of the outer rectangle is \(l + 3+3=l + 6\) and the width of the outer rectangle is \(w + 3+3=w + 6\).
The area of a rectangle is \(A=\text{length}\times\text{width}\). So the area of the outer rectangle \(A_{outer}=(l + 6)(w + 6)\).
Using the FOIL method: \((l + 6)(w + 6)=lw+6l + 6w+36\).

Step2: Find the area of the inner rectangle

The area of the inner rectangle (sitting area) is \(A_{inner}=lw\).

Step3: Find the area of the flower - border

The area of the flower - border \(A = A_{outer}-A_{inner}\).
Substitute the expressions for \(A_{outer}\) and \(A_{inner}\):
\(A=(lw + 6l+6w + 36)-lw\).
Simplify the expression: \(A=6l + 6w+36\).
Factor out the common factor of \(6\): \(A = 6(l + w+6)\).

Answer:

\(6(l + w + 6)\)