To find the volume of the dilated rectangular prism, we first need to calculate the volume of the original rectangular prism using the formula for volume:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]
Given the dimensions of the original rectangular prism:
- Width = 4 meters
- Length = 18 meters
- Height = 5 meters
Calculating the volume of the original prism:
\[ \text{Volume} = 18 , \text{m} \times 4 , \text{m} \times 5 , \text{m} = 360 , \text{m}^3 \]
When the prism is dilated by a scale factor of \(k\), the new volume is given by the formula:
\[ \text{New Volume} = k^3 \times \text{Original Volume} \]
Since the scale factor is 12:
\[ k = 12 \]
Now calculate the new volume:
\[ \text{New Volume} = 12^3 \times 360 , \text{m}^3 \]
Calculating \(12^3\):
\[ 12^3 = 1728 \]
Now, multiply by the original volume:
\[ \text{New Volume} = 1728 \times 360 , \text{m}^3 \]
Calculating \(1728 \times 360\):
\[ 1728 \times 360 = 622080 , \text{m}^3 \]
Therefore, the volume of the dilated rectangular prism is:
\[ \boxed{622080} , \text{m}^3 \]
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