To find the volume of the rectangular prism after dilation with a scale factor of 4, we first need to calculate the original volume of the prism.
The formula for the volume \( V \) of a rectangular prism is:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Given:
- Height = 2 meters
- Width = 3 meters
- Length = 4 meters
Calculating the original volume:
\[ V = 4 , \text{m} \times 3 , \text{m} \times 2 , \text{m} = 24 , \text{m}^3 \]
Now we apply the dilation. When a shape is dilated by a scale factor of \( k \), the new volume \( V' \) can be calculated by multiplying the original volume by \( k^3 \):
\[ V' = k^3 \times V \]
In this case, the scale factor \( k = 4 \):
\[ V' = 4^3 \times 24 , \text{m}^3 = 64 \times 24 , \text{m}^3 \]
Calculating \( 64 \times 24 \):
\[ 64 \times 24 = 1536 , \text{m}^3 \]
Thus, the volume of the rectangular prism after dilation is 1,536 m³.