To find the volume of the new rectangular prism after applying the scale factor, we first calculate the volume of the original rectangular prism using the dimensions provided:
Volume \( V \) of a rectangular prism is calculated using the formula: \[ V = \text{length} \times \text{width} \times \text{height} \]
For the original prism:
- Length = 5 cm
- Width = 6 cm
- Height = 4 cm
Calculating the volume: \[ V = 5 , \text{cm} \times 6 , \text{cm} \times 4 , \text{cm} = 120 , \text{cm}^3 \]
Now, since the scale factor is \( \frac{1}{2} \), we need to calculate the new volume using the scale factor's impact. When the dimensions are scaled down, the volume scales by the cube of the scale factor:
\[ \text{New Volume} = \text{Original Volume} \times \left(\text{scale factor}\right)^3 \] \[ \text{New Volume} = 120 , \text{cm}^3 \times \left(\frac{1}{2}\right)^3 = 120 , \text{cm}^3 \times \frac{1}{8} \] \[ \text{New Volume} = \frac{120}{8} = 15 , \text{cm}^3 \]
Thus, the volume of the new rectangular prism is \( \text{15 cm}^3 \).
The answer is: 15 cm³.