the length, width, and height of a rectangular prism can be represented by the expressions ((x + 3)), ((x + 7)) and ((x - 1)). write an expression to represent the surface area of the prism. ((sa = 2lw + 2lh + 2wh))

Question

Accepted Answer

# Explanation:
## Step1: Identify length, width, height
Let $ l = x + 3 $, $ w = x + 7 $, $ h = x - 1 $. The surface area formula is $ SA = 2lw + 2lh + 2wh $.

## Step2: Calculate $ 2lw $
$$
\begin{align*}
2lw &= 2(x + 3)(x + 7)\
&= 2(x^2 + 7x + 3x + 21)\
&= 2(x^2 + 10x + 21)\
&= 2x^2 + 20x + 42
\end{align*}
$$

## Step3: Calculate $ 2lh $
$$
\begin{align*}
2lh &= 2(x + 3)(x - 1)\
&= 2(x^2 - x + 3x - 3)\
&= 2(x^2 + 2x - 3)\
&= 2x^2 + 4x - 6
\end{align*}
$$

## Step4: Calculate $ 2wh $
$$
\begin{align*}
2wh &= 2(x + 7)(x - 1)\
&= 2(x^2 - x + 7x - 7)\
&= 2(x^2 + 6x - 7)\
&= 2x^2 + 12x - 14
\end{align*}
$$

## Step5: Sum the three terms
$$
\begin{align*}
SA &= (2x^2 + 20x + 42) + (2x^2 + 4x - 6) + (2x^2 + 12x - 14)\
&= (2x^2 + 2x^2 + 2x^2) + (20x + 4x + 12x) + (42 - 6 - 14)\
&= 6x^2 + 36x + 22
\end{align*}
$$

# Answer:
The expression for the surface area is $ 6x^2 + 36x + 22 $

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QUESTION IMAGE

Question

Explanation:

Step1: Identify length, width, height

Step2: Calculate \( 2lw \)

Step3: Calculate \( 2lh \)

Step4: Calculate \( 2wh \)

Step5: Sum the three terms

Answer: