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The width of a rectangular box is
2cm less than its length. The height of the box is
8cm. If the box has a length of
x centimeters, which of the following functions represents the surface area,
S, in square centimeters?
Choose 1 answer:
(A)
S=x^(2)+14 x-16
(B)
S=2x^(2)+28 x-32
(C)
S=16+2x-x^(2)
(D)
S=32+4x-2x^(2)

The width of a rectangular box is $2 \mathrm{~cm}$ less than its length. The height of the box is $8 \mathrm{~cm}$ . If the box has a length of $x$ centimeters, which of the following functions represents the surface area, $S$ , in square centimeters? $\newline$ Choose $1$ answer: $\newline$ (A) $S=x^{2}+14 x-16$ $\newline$ (B) $S=2 x^{2}+28 x-32$ $\newline$ (C) $S=16+2 x-x^{2}$ $\newline$ (D) $S=32+4 x-2 x^{2}$

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Solve a system of equations using substitution: word problems

Full solution

Q. The width of a rectangular box is

2 \mathrm{~cm}

less than its length. The height of the box is

8 \mathrm{~cm}

. If the box has a length of

x

centimeters, which of the following functions represents the surface area,

S

, in square centimeters?

\newline

Choose

1

answer:

\newline

(A)

S=x^{2}+14 x-16

\newline

(B)

S=2 x^{2}+28 x-32

\newline

(C)

S=16+2 x-x^{2}

\newline

(D)

S=32+4 x-2 x^{2}

Question prompt: The question prompt is: ＂What is the function that represents the surface area of the rectangular box in terms of its length $x$ ?＂
Define box dimensions: First, let's define the dimensions of the box using the given information. The length is $x$ cm, the width is $(x - 2)$ cm (since it's $2$ cm less than the length), and the height is $8$ cm.
Surface area formula: The surface area, $S$ , of a rectangular box is calculated by adding up the areas of all six faces. There are two faces for each pair of dimensions. The formula for the surface area is: $\newline$ $S = 2lw + 2lh + 2wh$ $\newline$ where $l$ is the length, $w$ is the width, and $h$ is the height.
Substitute dimensions: Substitute the given dimensions into the surface area formula: $\newline$ $S = 2(x)(x - 2) + 2(x)(8) + 2((x - 2)(8))$
Expand each term: Now, let's expand each term: $\newline$ S = $2$ (x^ $2$ - $2$ x) + $16$ x + $2$ ( $8$ x - $16$ )
Simplify equation: Simplify the equation by distributing and combining like terms: $\newline$ $S = 2x^2 - 4x + 16x + 16x - 32$
Combine x terms: Combine the x terms: $\newline$ $S = 2x^2 + 28x - 32$
Final function: The function that represents the surface area $S$ in terms of the length $x$ is: $\newline$ $S = 2x^2 + 28x - 32$ $\newline$ This corresponds to option $(B)$ .

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