To find the volume of a rectangular prism, we use the formula:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]
In the given prism, we have:
- Width \( W = 7 \) cm
- Length \( L = 6 \) cm
- Height \( H = 7 \) cm
First, we calculate the volume of the original prism:
\[ \text{Volume} = W \times L \times H = 7 \times 6 \times 7 \]
Calculating the above:
\[ 7 \times 6 = 42 \]
Then multiply that result by the height:
\[ 42 \times 7 = 294 \text{ cm}^3 \]
Now that we have the volume of the original prism, we need to find the volume of the new prism with a scale factor of \( \frac{1}{2} \). The volume scale factor is the cube of the linear scale factor. Therefore, we calculate it as follows:
\[ \text{Volume Scale Factor} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
Now, we multiply the volume of the original prism by the volume scale factor:
\[ \text{Volume of new prism} = \text{Original Volume} \times \text{Volume Scale Factor} \]
\[ \text{Volume of new prism} = 294 \times \frac{1}{8} = \frac{294}{8} = 36.75 \text{ cm}^3 \]
Thus, the volume of the new prism is:
\[ \text{Volume of new prism} = 36.75 \text{ cm}^3 \]
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