To find the new volume of the triangular prism after dilation with a scale factor of 3, you can follow these steps:
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Calculate the original volume of the triangular prism: The volume \( V \) of a triangular prism can be calculated using the formula: \[ V = \text{Base Area} \times \text{Height} \]
The base area of the triangular prism is: \[ \text{Base Area} = \frac{1}{2} \times \text{Base} \times \text{Height of Triangle} = \frac{1}{2} \times 7 , \text{m} \times 8 , \text{m} = 28 , \text{m}^2 \]
So the volume becomes: \[ V = \text{Base Area} \times \text{Length} = 28 , \text{m}^2 \times 12 , \text{m} = 336 , \text{m}^3 \]
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Dilation and Volume Scaling: When a prism is dilated by a scale factor of \( k \), the volume scales by a factor of \( k^3 \). Here, the scale factor \( k = 3 \).
Therefore, the new volume \( V' \) will be: \[ V' = V \times k^3 = 336 , \text{m}^3 \times 3^3 = 336 , \text{m}^3 \times 27 = 9072 , \text{m}^3 \]
So, the new volume of the triangular prism after dilation is 9,072 m³.