To determine the volume of a rectangular prism after dilation with a scale factor of 4, we need to consider how dilation affects volume.
When a three-dimensional shape is dilated by a scale factor \( k \), the new volume is given by the formula:
\[ \text{New Volume} = k^3 \times \text{Original Volume} \]
In this case, the scale factor \( k \) is 4. Therefore:
\[ k^3 = 4^3 = 64 \]
This means the new volume will be 64 times the original volume.
To find the original volume, we would need to either be given it directly or infer it from the list of possible answers. However, we don't have the original volume, but we can calculate what it might be if we look at the answer choices given, assuming they might represent the new volume after dilation.
If the new volume after dilation is one of the options provided, we can divide each option by 64 to see which one could potentially yield an integer volume for the original prism:
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For 1,536 m³: \[ \frac{1,536}{64} = 24 \quad \text{(possible original volume)} \]
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For 384 m³: \[ \frac{384}{64} = 6 \quad \text{(possible original volume)} \]
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For 6 m³: \[ \frac{6}{64} = 0.09375 \quad \text{(not an integer)} \]
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For 96 m³: \[ \frac{96}{64} = 1.5 \quad \text{(not an integer)} \]
From the calculations, the only potential new volumes that could correspond to integer original volumes are 1,536 m³ (which would correspond to an original volume of 24 m³) and 384 m³ (which would correspond to an original volume of 6 m³).
Since the question asks for the volume after dilation, we choose the largest option listed, which is:
1,536 m³.
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