The surface area \( S \) of a rectangular prism can be calculated using the formula:
\[ S = 2(lw + lh + wh) \]
where \( l \), \( w \), and \( h \) are the dimensions of the prism.
For the dimensions given:
- \( l = 334 , \text{dm} \)
- \( w = 412 , \text{dm} \)
- \( h = 5 , \text{dm} \)
Now, we will calculate the surface area step by step.
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Calculate \( lw \), \( lh \), and \( wh \):
\[ lw = 334 \times 412 = 137848 , \text{dm}^2 \]
\[ lh = 334 \times 5 = 1670 , \text{dm}^2 \]
\[ wh = 412 \times 5 = 2060 , \text{dm}^2 \]
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Sum these areas:
\[ lw + lh + wh = 137848 + 1670 + 2060 = 140578 , \text{dm}^2 \]
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Multiply by 2 to find the total surface area:
\[ S = 2 \times 140578 = 281156 , \text{dm}^2 \]
Given the response options and our calculations, none of the provided choices match the calculated surface area. It seems there may have been an error in the options provided, or the correct surface area is not listed. Based on our computation, the surface area of the rectangular prism is:
\[ 281156 , \text{dm}^2 \]