To find the volume of the new rectangular prism after applying a scale factor of \( \frac{1}{2} \), we first need to calculate the volume of the original rectangular prism.
The volume \( V \) of a rectangular prism is given by the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
In this case, the dimensions of the original rectangular prism are:
- Length = 6 cm
- Width = 5 cm
- Height = 4 cm
Calculating the volume:
\[ V = 6 , \text{cm} \times 5 , \text{cm} \times 4 , \text{cm} = 120 , \text{cm}^3 \]
Now, since the new rectangular prism has a scale factor of \( \frac{1}{2} \):
- The volume of a prism changes with the cube of the scale factor. So we need to calculate the new volume as follows:
\[ \text{New Volume} = \text{Original Volume} \times \left( \text{Scale Factor} \right)^3 \]
Letting \( \text{Scale Factor} = \frac{1}{2} \):
\[ \text{New Volume} = 120 , \text{cm}^3 \times \left( \frac{1}{2} \right)^3 \]
Calculating \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \):
\[ \text{New Volume} = 120 , \text{cm}^3 \times \frac{1}{8} = \frac{120}{8} , \text{cm}^3 = 15 , \text{cm}^3 \]
Thus, the volume of the new rectangular prism is:
\[ \boxed{15 , \text{cm}^3} \]
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