Hector, a rug maker, is working on a custom rectangular rug for a client's living room. To fit the room, the rug needs to be $2$ times as long as it is wide. In addition to the interior pattern, which Hector will design, the rug will have a floral border that is $1$ foot wide on each side. $\newline$ The area of the interior pattern of the rug in square feet can be modeled by the expression $(2x - 2)(x - 2)$ , where $x$ is the width of the entire rug in feet. $\newline$ What does the quantity $2x - 2$ represent in the expression? $\newline$ Choices: $\newline$ (A)the area of the border in square feet $\newline$ (B)the perimeter around the interior pattern of the rug in feet $\newline$ (C)the area of the interior pattern of the rug in square feet $\newline$ (D)the length of the interior pattern of the rug in feet

Full solution

Q. Hector, a rug maker, is working on a custom rectangular rug for a client's living room. To fit the room, the rug needs to be

2

times as long as it is wide. In addition to the interior pattern, which Hector will design, the rug will have a floral border that is

1

foot wide on each side.

\newline

The area of the interior pattern of the rug in square feet can be modeled by the expression

(2x - 2)(x - 2)

, where

x

is the width of the entire rug in feet.

\newline

What does the quantity

2x - 2

represent in the expression?

\newline

Choices:

\newline

(A)the area of the border in square feet

\newline

(B)the perimeter around the interior pattern of the rug in feet

\newline

(C)the area of the interior pattern of the rug in square feet

\newline

(D)the length of the interior pattern of the rug in feet

Expression for Area: The expression for the area of the interior pattern is $(2x - 2)(x - 2)$ . We know that $x$ is the width of the entire rug, including the border.
Width of Interior Pattern: Since the border is $1$ foot wide on each side, we subtract $2$ feet from the width to get the width of the interior pattern, which is $x - 2$ .
Length Calculation: The rug needs to be $2$ times as long as it is wide, so the length of the entire rug is $2x$ . To find the length of the interior pattern, we subtract the border's width from each side, which is $2$ feet in total, giving us $2x - 2$ .
Representation of Length: Therefore, $2x - 2$ represents the length of the interior pattern of the rug in feet, not the area of the border, the perimeter, or the area of the interior pattern.