To determine the volume of a rectangular prism, you can use the formula:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]
In this case, the dimensions of the rectangular prism are given as 4, 3, and 2. Thus, the initial volume of the prism can be calculated as follows:
\[ \text{Volume} = 4 \times 3 \times 2 = 24 \]
Now, if the prism is dilated using a scale factor of 4, each dimension will be multiplied by that scale factor. Therefore, the new dimensions will be:
\[ \text{New length} = 4 \times 4 = 16 \] \[ \text{New width} = 3 \times 4 = 12 \] \[ \text{New height} = 2 \times 4 = 8 \]
Now, calculate the volume of the dilated prism:
\[ \text{New Volume} = \text{New length} \times \text{New width} \times \text{New height} \] \[ \text{New Volume} = 16 \times 12 \times 8 \]
Calculating step by step:
- \( 16 \times 12 = 192 \)
- \( 192 \times 8 = 1536 \)
Thus, the volume of the rectangular prism after dilation is:
\[ \text{New Volume} = 1536 \]
Alternatively, there is a shortcut for calculating the volume change due to dilation. The volume of a shape scales by the cube of the scale factor. Since the scale factor is 4, the volume increases by:
\[ \text{New Volume} = \text{Old Volume} \times (4^3) = 24 \times 64 = 1536 \]
Either way, the final volume of the dilated rectangular prism is:
\[ \boxed{1536} \]