To find the volume of a rectangular prism, you can use the formula:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
For the original prism, we know:
- Width (W) = 7 cm
- Length (L) = 6 cm
- Height (H) = 4 cm
So, the volume of the original prism is:
\[ \text{Volume} = 7 , \text{cm} \times 6 , \text{cm} \times 4 , \text{cm} = 168 , \text{cm}^3 \]
Since the new prism is similar to the original one and has a scale factor of \( \frac{1}{2} \), the dimensions of the new prism will be:
- New Width = \( 7 \times \frac{1}{2} = 3.5 \) cm
- New Length = \( 6 \times \frac{1}{2} = 3 \) cm
- New Height = \( 4 \times \frac{1}{2} = 2 \) cm
Now, we can calculate the volume of the new prism using the new dimensions:
\[ \text{New Volume} = 3.5 , \text{cm} \times 3 , \text{cm} \times 2 , \text{cm} \]
Calculating this gives:
\[ \text{New Volume} = 3.5 \times 3 \times 2 = 21 , \text{cm}^3 \]
Alternatively, you can determine the volume of the new prism based on the scale factor of the volume due to similarity. The volume of similar solids scales by the cube of the scale factor:
\[ \text{Scale Factor (Volume)} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
Thus, the volume of the new prism can also be calculated as:
\[ \text{New Volume} = 168 , \text{cm}^3 \times \frac{1}{8} = 21 , \text{cm}^3 \]
Therefore, the volume of the new prism is:
\[ \boxed{21 , \text{cm}^3} \]