To find the new volume of the triangular prism after dilation, we first need to calculate the volume of the original prism and then apply the scale factor to that volume.
The formula for the volume \( V \) of a triangular prism is:
\[ V = \text{Area of base} \times \text{Height} \]
- First, we need to determine the area of the triangular base. The base is triangular and has a width of 7 meters and a height of 8 meters. The area \( A \) of the triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \times 8 = 28 , \text{m}^2 \]
- The height of the prism is the length of 12 meters. So, we calculate the volume of the original prism:
\[ V_{\text{original}} = A \times \text{Length} = 28 \times 12 = 336 , \text{m}^3 \]
- When the prism is dilated with a scale factor of 3, the new volume \( V_{\text{new}} \) can be calculated by cubing the scale factor and multiplying it by the original volume:
\[ V_{\text{new}} = V_{\text{original}} \times \text{(scale factor)}^3 = 336 \times 3^3 = 336 \times 27 = 9072 , \text{m}^3 \]
Thus, the new volume of the dilated triangular prism is 9,072 m³.
So the answer is 9,072 m3.