To determine the surface area of a rectangular prism with dimensions [tex]\( x \)[/tex], [tex]\( 2x \)[/tex], and [tex]\( x+8 \)[/tex], we need to use the formula for the surface area of a rectangular prism:
[tex]\[ \text{Surface Area} = 2(lw + lh + wh) \][/tex]
where:
- [tex]\( l \)[/tex] is the length,
- [tex]\( w \)[/tex] is the width,
- [tex]\( h \)[/tex] is the height.
Given:
- [tex]\( l = x \)[/tex],
- [tex]\( w = 2x \)[/tex],
- [tex]\( h = x + 8 \)[/tex].
First, calculate the areas of the three distinct pairs of faces:
1. Area of the face with dimensions [tex]\( l \)[/tex] and [tex]\( w \)[/tex]:
[tex]\[ lw = x \cdot 2x = 2x^2 \][/tex]
2. Area of the face with dimensions [tex]\( l \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ lh = x \cdot (x + 8) = x(x + 8) = x^2 + 8x \][/tex]
3. Area of the face with dimensions [tex]\( w \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ wh = 2x \cdot (x + 8) = 2x(x + 8) = 2x^2 + 16x \][/tex]
Next, add these areas together:
[tex]\[ lw + lh + wh = 2x^2 + (x^2 + 8x) + (2x^2 + 16x) \][/tex]
Combine like terms:
[tex]\[ 2x^2 + x^2 + 8x + 2x^2 + 16x = (2x^2 + x^2 + 2x^2) + (8x + 16x) \][/tex]
[tex]\[ = 5x^2 + 24x \][/tex]
Now, multiply by 2 to account for both sets of faces:
[tex]\[ \text{Surface Area} = 2(lw + lh + wh) = 2(5x^2 + 24x) \][/tex]
[tex]\[ = 10x^2 + 48x \][/tex]
Thus, the expression that represents the surface area of the rectangular prism is:
[tex]\[ 10x^2 + 48x \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{10x^2 + 48x} \][/tex] square units.
