To determine the volume of the rectangular prism and then find the volume after dilating it with a scale factor of 4, we first calculate the original volume.
The formula for the volume \( V \) of a rectangular prism is: \[ V = \text{length} \times \text{width} \times \text{height} \]
Given:
- Height = 2 m
- Width = 8 m
- Length = 4 m
Let's calculate the original volume:
\[ V = 4 , \text{m} \times 8 , \text{m} \times 2 , \text{m} = 64 , \text{m}^3 \]
Next, when a shape is dilated by a scale factor \( k \), the new volume \( V' \) is given by: \[ V' = k^3 \times V \] In this case, the scale factor \( k = 4 \).
So we calculate: \[ V' = 4^3 \times 64 , \text{m}^3 \] Calculating \( 4^3 \): \[ 4^3 = 64 \] Now, substitute this back into the volume formula: \[ V' = 64 \times 64 = 4096 , \text{m}^3 \]
However, my calculations above are incorrect. The volume after dilation should be:
- Original Volume \( = 64 , \text{m}^3 \)
- Scale Factor \( = 4 \), so \( 4^3 = 64 \)
Hence: \[ V' = 64 \times 64 = 4096 \text{ m}^3 \]
Therefore, the volume of the rectangular prism after dilation is 4096 m³.
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