To find the volume of a triangular prism, we first use the formula for the volume \( V \):
\[ V = \text{Base Area} \times \text{Height} \]
For a triangular prism, the base is a triangle. The area \( A \) of the triangular base can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, let's assume the triangular base has a width of 8 cm and a height of 6 cm. Thus, the base area is:
\[ A = \frac{1}{2} \times 8 , \text{cm} \times 6 , \text{cm} = \frac{1}{2} \times 48 , \text{cm}^2 = 24 , \text{cm}^2 \]
Now to find the volume of the triangular prism:
\[ V = A \times \text{length} = 24 , \text{cm}^2 \times 10 , \text{cm} = 240 , \text{cm}^3 \]
Next, if the figure is dilated using a scale factor of 2, the dimensions of the prism will be scaled as follows:
- New width = \( 8 , \text{cm} \times 2 = 16 , \text{cm} \)
- New height = \( 6 , \text{cm} \times 2 = 12 , \text{cm} \)
- New length = \( 10 , \text{cm} \times 2 = 20 , \text{cm} \)
We now find the new volume using the new dimensions. The base area of the dilated triangular prism becomes:
\[ A' = \frac{1}{2} \times 16 , \text{cm} \times 12 , \text{cm} = \frac{1}{2} \times 192 , \text{cm}^2 = 96 , \text{cm}^2 \]
Thus, the new volume \( V' \) is:
\[ V' = A' \times \text{new length} = 96 , \text{cm}^2 \times 20 , \text{cm} = 1920 , \text{cm}^3 \]
Therefore, the volume of the dilated triangular prism is:
\[ \boxed{1920} , \text{cm}^3 \]
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