Blue-Green Lagoons is a company that designs and installs wading pools. The standard design includes a rectangular pool that is $2$ times as long as it is wide. The wading pool has an $8$ -foot-wide patio on all sides. $\newline$ The total area of the pool and patio in square feet can be modeled by the expression $(2x + 16)(x + 16)$ , where $x$ is the width of the pool in feet. This expression can also be written in the form $2x^2 + 48x + 256$ . $\newline$ What does the quantity $48x + 256$ represent in the expression? $\newline$ Choices: $\newline$ (A)the area of the patio in square feet $\newline$ (B)the length of the pool and patio together in feet $\newline$ (C)the perimeter around the outside of the patio in feet $\newline$ (D)the area of the pool in square feet

Full solution

Q. Blue-Green Lagoons is a company that designs and installs wading pools. The standard design includes a rectangular pool that is

2

times as long as it is wide. The wading pool has an

8

-foot-wide patio on all sides.

\newline

The total area of the pool and patio in square feet can be modeled by the expression

(2x + 16)(x + 16)

, where

x

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

2x^2 + 48x + 256

\newline

What does the quantity

48x + 256

represent in the expression?

\newline

Choices:

\newline

(A)the area of the patio in square feet

\newline

(B)the length of the pool and patio together in feet

\newline

(C)the perimeter around the outside of the patio in feet

\newline

(D)the area of the pool in square feet

Total Area Expression: The expression for the total area is \(2x + $16$ )x + \(16)\, which represents the area of the pool and the patio together.
Expanding Expression: Expanding the expression, we get $2x^2 + 32x + 16x + 256$ , which simplifies to $2x^2 + 48x + 256$ .
Pool Area Calculation: The term $2x^2$ represents the area of the pool, since the pool is $2$ times as long as it is wide, so its area is length times width, or $2x \times x$ .
Patio Area Calculation: The term $48x + 256$ must then represent the area of the patio, since it's the remaining part of the expression for the total area after accounting for the pool's area.