Question

A square rug has an inner square in the center. The side length of inner square is x inches and the width of the outer region is 2 inches. What is the area of the outer part of the rug?

Answer 1

To find the area of the outer part of the rug, we need to follow these steps:

1. Calculate the area of the entire square rug including the inner square.
2. Calculate the area of the inner square.
3. Subtract the area of the inner square from the area of the entire rug to find the area of the outer part.

The side length of the inner square is x inches.

Since the width of the outer region is 2 inches on all sides, this width will be added to each side of the inner square twice (once for each side of the corner) when calculating the side length of the entire rug. Thus, the entire side length of the rug will be x + 2 + 2 inches, which simplifies to x + 4 inches.

Now let's perform the calculations step-wise:

Step 1: Calculate the area of the entire square rug.
The formula for the area of a square is side length squared (A = side^2), so the area of the entire rug is (x + 4)^2 square inches.

Step 2: Calculate the area of the inner square.
Using the same formula (A = side^2), the area of the inner square is x^2 square inches.

Step 3: Subtract the area of the inner square from the area of the entire rug.
Now we subtract the area of the inner square from the area of the entire rug to find the area of the outer region:
Area of the outer part = Area of the entire rug - Area of the inner square
Area of the outer part = (x + 4)^2 - x^2

To make it clearer, let's expand (x + 4)^2 using the distributive property (FOIL: First, Outer, Inner, Last):

(x + 4)^2 = (x + 4)(x + 4)
= x*x + 4*x + 4*x + 4*4
= x^2 + 4x + 4x + 16
= x^2 + 8x + 16

So the Area of the outer part is:

Area of the outer part = x^2 + 8x + 16 - x^2
Area of the outer part = 8x + 16 square inches

This is the area of the outer part of the rug.

Answer 2

so Area of Outer part = 8x + 16 square inches

Explanation:

2 squares.

inner square dimension is x inches by x inches.

width of outer region= 2inches.

We want the area of the border around the inner square.

so

Area of border = B = Area of Large - Area of Inner

B = (x+4)^2 - x^2

B = x^2 + 8x + 16 - x^2

B = 8x + 16 square inches

so Area of Outer part = 8x + 16 square inches