When a three-dimensional shape, such as a triangular prism, is dilated using a scale factor (k), the volume of the shape changes by a factor of \( k^3 \).
If the original volume of the triangular prism is \( V \), then after dilation with a scale factor of \( k = 3 \), the new volume \( V' \) can be calculated as follows:
\[ V' = k^3 \times V = 3^3 \times V = 27 \times V \]
To determine the new volume, we must know the original volume \( V \). Unfortunately, the original volume isn't given in your question.
However, if we assume that the options provided, such as 112 m³, 9,072 m³, 1,008 m³, and 27,216 m³, are the possible new volumes after dilation, we can work backwards to find the original volume for each option:
For each possible new volume \( V' \):
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112 m³: \[ V = \frac{112}{27} \approx 4.15 , \text{m}^3 \]
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9,072 m³: \[ V = \frac{9072}{27} = 336 , \text{m}^3 \]
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1,008 m³: \[ V = \frac{1008}{27} \approx 37.33 , \text{m}^3 \]
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27,216 m³: \[ V = \frac{27216}{27} = 1008 , \text{m}^3 \]
Out of these calculations, if we were to multiply the original corresponding volumes by 27 to check which matches the normal range of volumes (most likely candidates), 9,072 m³ and 27,216 m³ stand out here.
However, without the exact original volume of the triangular prism, we recognize that if we focus on the provided options, the answer that would align with the dilation of a reasonable prism is:
27,216 m³.
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