Question

1. a garden is surrounded by a paved border of uniform width (see the diagram below). write an expression for the area of the border. the dimensions are given in m.

Explanation:

Step1: Find the area of the larger rectangle

The length of the larger rectangle is \((13 + 2x)\)m and the width is \((11+ 2x)\)m. Using the formula for the area of a rectangle \(A = length\times width\), the area of the larger rectangle is \(A_{1}=(13 + 2x)(11+ 2x)\).
Expand \((13 + 2x)(11+ 2x)\) using the FOIL method:

$$
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$$

Step2: Find the area of the smaller rectangle

The length of the smaller rectangle is \(13\)m and the width is \(11\)m. Using the formula \(A = length\times width\), the area of the smaller rectangle is \(A_{2}=13\times11 = 143\) \(m^{2}\).

Step3: Find the area of the border

The area of the border \(A\) is the difference between the area of the larger rectangle and the area of the smaller rectangle.
\(A=A_{1}-A_{2}=(4x^{2}+48x + 143)-143\)
Simplify the expression: \(A = 4x^{2}+48x\)

Answer:

The area of the border is \(4x^{2}+48x\) \(m^{2}\)