To find the volume of a triangular prism, you can use the formula:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
The base area for a triangular prism can be calculated using the formula for the area of a triangle:
\[ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, let's assume the "width" is the base of the triangle and the "height" is the height of the triangle.
Given:
- Length of the prism = 12 m
- Width (base of the triangle) = 7 m
- Height of the triangle = 8 m
First, calculate the area of the triangular base:
\[ \text{Base Area} = \frac{1}{2} \times 7 , \text{m} \times 8 , \text{m} = \frac{1}{2} \times 56 , \text{m}^2 = 28 , \text{m}^2 \]
Next, calculate the volume of the prism using the length:
\[ \text{Volume} = \text{Base Area} \times \text{Length} = 28 , \text{m}^2 \times 12 , \text{m} = 336 , \text{m}^3 \]
When the prism is dilated using a scale factor of 3, each dimension (length, width, height) is multiplied by 3. Therefore, the new dimensions will be:
- New length = \( 12 , \text{m} \times 3 = 36 , \text{m} \)
- New width (base of triangle) = \( 7 , \text{m} \times 3 = 21 , \text{m} \)
- New height of the triangle = \( 8 , \text{m} \times 3 = 24 , \text{m} \)
Now, we calculate the new base area:
\[ \text{New Base Area} = \frac{1}{2} \times 21 , \text{m} \times 24 , \text{m} = \frac{1}{2} \times 504 , \text{m}^2 = 252 , \text{m}^2 \]
Now calculate the new volume of the prism:
\[ \text{New Volume} = \text{New Base Area} \times \text{New Length} = 252 , \text{m}^2 \times 36 , \text{m} = 9072 , \text{m}^3 \]
Thus, the new volume of the dilated triangular prism is:
\[ \boxed{9072 , \text{m}^3} \]
Alternatively, since volume scales by the cube of the scale factor, you can also calculate it simply by multiplying the original volume by \( 3^3 \):
\[ \text{New Volume} = 336 , \text{m}^3 \times 27 = 9072 , \text{m}^3 \]
This confirms our previous calculation.