To find the original volume of the triangular prism, we use the formula for the volume of a triangular prism:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
First, we need to find the base area of the triangular base. The prism has a width of 7 meters (base of the triangle) and a height of 8 meters (height of the triangle). The area of the triangle can be calculated as:
\[ \text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 , \text{m} \times 8 , \text{m} = 28 , \text{m}^2 \]
Now we calculate the volume of the prism with the length of 12 meters:
\[ \text{Volume} = \text{Base Area} \times \text{Length} = 28 , \text{m}^2 \times 12 , \text{m} = 336 , \text{m}^3 \]
Next, we need to apply the dilation with a scale factor of 3. When a shape is dilated by a scale factor \( k \), the volume scales by the cube of that scale factor:
\[ \text{New Volume} = \text{Original Volume} \times k^3 = 336 , \text{m}^3 \times 3^3 = 336 , \text{m}^3 \times 27 \]
Calculating that gives:
\[ \text{New Volume} = 336 \times 27 = 9072 , \text{m}^3 \]
Thus, the new volume of the dilated triangular prism is 9,072 m³.