To find the volume of a triangular prism, you can use the formula:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
For a triangular prism, the base area (A) is calculated as:
\[ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height (of triangle)} \]
However, based on the dimensions provided (width, length, and height of the prism), let's interpret the width as the base of the triangle and the height of the triangular face as what we need to define:
Assume:
- Width = Base of triangle = 7 meters
- Length = Length of the prism = 12 meters
- Height = The height of the triangular face = 8 meters
Calculating the volume of the triangular prism:
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Calculate the base area: \[ A = \frac{1}{2} \times 7 \times 8 = \frac{56}{2} = 28 \text{ m}^2 \]
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Now calculate the volume: \[ \text{Volume} = \text{Base Area} \times \text{Length} = 28 \text{ m}^2 \times 12 \text{ m} = 336 \text{ m}^3 \]
Now, if the prism is dilated using a scale factor of 3, the volume of the dilated prism will be impacted by the cube of the scale factor. Thus, the new volume can be calculated as follows:
\[ \text{New Volume} = \text{Original Volume} \times (\text{Scale Factor})^3 \] \[ \text{New Volume} = 336 \text{ m}^3 \times 3^3 = 336 \text{ m}^3 \times 27 = 9072 \text{ m}^3 \]
Therefore, the new volume of the dilated triangular prism is:
9,072 m³
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