To determine the volume of the rectangular prism after dilation, we first calculate the original volume and then apply the scale factor.
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Calculate the original volume of the rectangular prism: \[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \] \[ \text{Volume} = 4 , \text{m} \times 3 , \text{m} \times 2 , \text{m} = 24 , \text{m}^3 \]
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Determine the new dimensions after dilation: With a scale factor of 4, the new dimensions of the prism will be:
- New width = \( 3 \times 4 = 12 \) meters
- New length = \( 4 \times 4 = 16 \) meters
- New height = \( 2 \times 4 = 8 \) meters
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Calculate the volume of the dilated prism: \[ \text{New Volume} = \text{new length} \times \text{new width} \times \text{new height} \] \[ \text{New Volume} = 16 , \text{m} \times 12 , \text{m} \times 8 , \text{m} = 1536 , \text{m}^3 \]
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Conclusion: The volume of the rectangular prism after being dilated using a scale factor of 4 is \( 1,536 , \text{m}^3 \).
The correct response is: 1,536 m³