To find the volume of the new rectangular prism with a scale factor of 12, we first need to calculate the volume of the original rectangular prism, and then apply the scale factor.
The volume \( V \) of a rectangular prism is calculated using the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Given the dimensions of the original rectangular prism:
- Width = 6 cm
- Length = 5 cm
- Height = 4 cm
We can calculate the volume of the original rectangular prism:
\[ V = 6 , \text{cm} \times 5 , \text{cm} \times 4 , \text{cm} = 120 , \text{cm}^3 \]
When we apply the scale factor of 12, the volume of the new rectangular prism increases by the cube of the scale factor. The formula for the volume of a similar shape with a scale factor \( k \) is:
\[ V' = k^3 \times V \]
Plugging in the values:
\[ V' = 12^3 \times 120 , \text{cm}^3 \]
Calculating \( 12^3 \):
\[ 12^3 = 1728 \]
Now, we can find the volume of the new rectangular prism:
\[ V' = 1728 \times 120 , \text{cm}^3 \]
Calculating this:
\[ V' = 207360 , \text{cm}^3 \]
Thus, the volume of the new rectangular prism is \( 207360 , \text{cm}^3 \).
However, if we only consider the initial calculation of the original volumes, it is clear that none of the available response options are correct based on the calculation of \( 207360 , \text{cm}^3 \).
If you have additional context stating otherwise or it requires to choose the best-match among the options provided, please provide clarification.