To determine 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 volume \( V \) of a rectangular prism can be calculated using the formula:
\[ V = L \times W \times H \]
Where:
- \( L \) = Length = 4 m
- \( W \) = Width = 3 m
- \( H \) = Height = 2 m
Calculating the original volume:
\[ V = 4 , \text{m} \times 3 , \text{m} \times 2 , \text{m} = 24 , \text{m}^3 \]
Now, when the prism is dilated using a scale factor of 4, the new dimensions will be:
- New length \( L' = 4 \times 4 = 16 , \text{m} \)
- New width \( W' = 4 \times 3 = 12 , \text{m} \)
- New height \( H' = 4 \times 2 = 8 , \text{m} \)
Now, we calculate the new volume \( V' \) using the new dimensions:
\[ V' = L' \times W' \times H' = 16 , \text{m} \times 12 , \text{m} \times 8 , \text{m} \]
Calculating this step-by-step:
- Calculate \( 16 , \text{m} \times 12 , \text{m} = 192 , \text{m}^2 \)
- Then calculate \( 192 , \text{m}^2 \times 8 , \text{m} = 1536 , \text{m}^3 \)
Thus, the volume of the dilated rectangular prism is:
\[ \boxed{1536 , \text{m}^3} \]