To find the volume of the rectangular prism before dilation, use the formula for the volume of a rectangular prism:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
Given the dimensions:
- Length (L) = 4 m
- Width (W) = 3 m
- Height (H) = 2 m
The original volume is calculated as follows:
\[ \text{Volume} = 4 , \text{m} \times 3 , \text{m} \times 2 , \text{m} = 24 , \text{m}^3 \]
Next, when the prism is dilated using a scale factor of 4, the new dimensions will be:
- New Length = \(4 \times 4 = 16 , \text{m}\)
- New Width = \(4 \times 3 = 12 , \text{m}\)
- New Height = \(4 \times 2 = 8 , \text{m}\)
Now, calculate the volume of the dilated rectangular prism:
\[ \text{New Volume} = 16 , \text{m} \times 12 , \text{m} \times 8 , \text{m} \]
Calculating this gives:
\[ \text{New Volume} = 16 \times 12 = 192 , \text{m}^2 \] \[ \text{New Volume} = 192 \times 8 = 1536 , \text{m}^3 \]
Thus, the volume of the rectangular prism after dilation is:
\[ \boxed{1536 , \text{m}^3} \]
Alternatively, you can find the new volume by using the property of dilations in three dimensions:
The volume scales by the cube of the scale factor. Therefore, with a scale factor of 4:
\[ \text{New Volume} = \text{Original Volume} \times \text{Scale Factor}^3 \] \[ \text{New Volume} = 24 , \text{m}^3 \times 4^3 = 24 , \text{m}^3 \times 64 = 1536 , \text{m}^3 \]
Both methods confirm that the new volume is:
\[ \boxed{1536 , \text{m}^3} \]