Tim has decided to add a wraparound porch to his beach cottage so that he can spend more time outside. His cottage is $45$ feet long and $25$ feet wide. The porch will go around three sides of the cottage and be the same width on all sides. $\newline$ The total area of the cottage and porch together in square feet can be modeled by the expression $(45 + 2x)(25 + x)$ , where $x$ is the width of the porch in feet. This expression can be also written in the form $2x^2 + 95x + 1,125$ . $\newline$ What does the quantity $2x^2 + 95x$ represent in the expression? $\newline$ Choices: $\newline$ (A)the perimeter around the outside of the porch in feet $\newline$ (B)the area of the cottage in square feet $\newline$ (C)the area of the porch in square feet $\newline$ (D)the length of the cottage and porch together in feet

Full solution

Q. Tim has decided to add a wraparound porch to his beach cottage so that he can spend more time outside. His cottage is

45

feet long and

25

feet wide. The porch will go around three sides of the cottage and be the same width on all sides.

\newline

The total area of the cottage and porch together in square feet can be modeled by the expression

(45 + 2x)(25 + x)

, where

x

is the width of the porch in feet. This expression can be also written in the form

2x^2 + 95x + 1,125

\newline

What does the quantity

2x^2 + 95x

represent in the expression?

\newline

Choices:

\newline

(A)the perimeter around the outside of the porch in feet

\newline

(B)the area of the cottage in square feet

\newline

(C)the area of the porch in square feet

\newline

(D)the length of the cottage and porch together in feet

Expand Expression: The expression for the total area is $(45 + 2x)(25 + x)$ . Let's expand this to see what each term represents.
Calculate Cottage Area: First, multiply $45$ by $25$ to get the area of the cottage without the porch. $\newline$ $45 \times 25 = 1125$ square feet.
Calculate Porch Areas: Next, multiply $45$ by $x$ and $25$ by $2x$ to get the area of the porch on the long side and the short side, respectively. $\newline$ $45x + 50x = 95x$ square feet.
Determine Total Area: Finally, multiply $2x$ by $x$ to get the area of the porch on the corner, which wraps around twice. $\newline$ $2x \times x = 2x^2$ square feet.
Identify Porch Area: Now, we have the expanded form: $2x^2 + 95x + 1125$ . The $1125$ represents the area of the cottage, and the terms $2x^2 + 95x$ represent the area of the porch.
Identify Porch Area: Now, we have the expanded form: $2x^2 + 95x + 1125$ . The $1125$ represents the area of the cottage, and the terms $2x^2 + 95x$ represent the area of the porch.So, the quantity $2x^2 + 95x$ represents the area of the porch in square feet.