Question

a square rug has an inner square in the center. the side length of the inner square is x inches and the width of the outer region is 12 in. what is the area of the outer part of the rug?

Explanation:

Step1: Find side - length of outer square

The side - length of the outer square is the sum of the side - length of the inner square and twice the width of the outer region. So the side - length of the outer square $s=x + 2\times12=x + 24$ inches.

Step2: Find area of outer square

The area formula for a square is $A=s^{2}$. Substituting $s=x + 24$ into the formula, we get $A=(x + 24)^{2}$. Using the formula $(a + b)^{2}=a^{2}+2ab + b^{2}$, where $a=x$ and $b = 24$, we have $A=x^{2}+48x+576$ square inches.

Step3: Find area of inner square

The area of the inner square with side - length $x$ is $A_{inner}=x^{2}$ square inches.

Step4: Find area of outer part

The area of the outer part of the rug is the area of the outer square minus the area of the inner square. So $A_{outer}=x^{2}+48x + 576-x^{2}=48x+576$ square inches.

Answer:

$48x + 576$ square inches