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A rectangular prism has dimensions [tex]$x$[/tex] units, [tex]$2x$[/tex] units, and [tex][tex]$x+8$[/tex][/tex] units. Which expression represents the surface area of the prism?

A. [tex]8x + 16[/tex] square units
B. [tex]16x + 32[/tex] square units
C. [tex]10x^2 + 48x[/tex] square units
D. [tex]2x^3 + 16x^2[/tex] square units


Sagot :

To find the surface area of a rectangular prism with dimensions [tex]\(x\)[/tex] units, [tex]\(2x\)[/tex] units, and [tex]\(x + 8\)[/tex] units, let's follow these steps:

1. Identify the dimensions:
- Length ([tex]\(l\)[/tex]): [tex]\(x\)[/tex] units
- Width ([tex]\(w\)[/tex]): [tex]\(2x\)[/tex] units
- Height ([tex]\(h\)[/tex]): [tex]\(x + 8\)[/tex] units

2. Surface area of a rectangular prism:
The formula for the surface area ([tex]\(SA\)[/tex]) of a rectangular prism is:
[tex]\[ SA = 2(lw + lh + wh) \][/tex]

3. Plug in the dimensions into the formula:
[tex]\[ SA = 2(x \cdot 2x + x \cdot (x + 8) + 2x \cdot (x + 8)) \][/tex]

4. Calculate each part of the expression inside the parentheses:
- [tex]\(lw = x \cdot 2x = 2x^2\)[/tex]
- [tex]\(lh = x \cdot (x + 8) = x^2 + 8x\)[/tex]
- [tex]\(wh = 2x \cdot (x + 8) = 2x^2 + 16x\)[/tex]

5. Add these parts together:
[tex]\[ 2x^2 + x^2 + 8x + 2x^2 + 16x = 5x^2 + 24x \][/tex]

6. Multiplying by 2 to get the total surface area:
[tex]\[ SA = 2 \cdot (5x^2 + 24x) = 10x^2 + 48x \][/tex]

Thus, the expression that represents the surface area of the prism is:
[tex]\[ 10x^2 + 48x \text{ square units} \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{10 x^2 + 48 x \text{ square units}} \][/tex]