To find the volume of the new rectangular prism, we first need to calculate the volume of the original rectangular prism. The volume \( V \) of a rectangular prism is given by the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
For the original prism with dimensions:
- Length = 5 cm
- Width = 6 cm
- Height = 4 cm
The volume is calculated as follows:
\[ V = 5 , \text{cm} \times 6 , \text{cm} \times 4 , \text{cm} = 120 , \text{cm}^3 \]
Now, since the new rectangular prism is similar to the original one with a scale factor of 12, the volume of similar shapes changes according to the cube of the scale factor.
Thus, the volume of the new rectangular prism can be calculated using the following formula:
\[ V_{\text{new}} = V_{\text{original}} \times \text{scale factor}^3 \]
Substituting in the values:
\[ V_{\text{new}} = 120 , \text{cm}^3 \times 12^3 \]
Calculating \( 12^3 \):
\[ 12^3 = 1728 \]
Now we multiply that by the original volume:
\[ V_{\text{new}} = 120 , \text{cm}^3 \times 1728 = 207360 , \text{cm}^3 \]
Since this is not one of the options provided in the responses, let's revisit the response choices:
- 30 cm³
- 15 cm³
- 60 cm³
- 120 cm³
These choices likely correspond to scaled measurements not being the new volume. Thus, if you actually just meant the original volume notation for simplified options without scaling factors context, please stick to understanding the original queries while expanding understanding relating scale factors and volumes appropriately.
If strictly by options suggested without broader context or scale application, the closest interpretation returns to original value or evaluation context at 120 cm³ before considered scale adjustments or novel positioning against final metrics. Would require more detail on the relevance behind scale and initial context from scaling upwards.