To find the new volume of the triangular prism after dilation, we first need to calculate the volume of the original triangular prism and then apply the scale factor.
The volume \( V \) of a triangular prism can be calculated using the formula: \[ V = \text{Base Area} \times \text{Height} \]
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Calculate the area of the triangular base: The base of the prism has a width of 7 meters and a height of 8 meters. Using the formula for the area of a triangle (\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)), we can calculate the area: \[ \text{Area} = \frac{1}{2} \times 7 \times 8 = \frac{1}{2} \times 56 = 28 \text{ m}^2 \]
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Calculate the volume of the original prism: The prism has a length of 12 meters, so we can plug the area into the volume formula: \[ V = \text{Base Area} \times \text{Length} = 28 \text{ m}^2 \times 12 \text{ m} = 336 \text{ m}^3 \]
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Calculate the new volume after dilation: When the prism is dilated using a scale factor of 3, the volume is affected by the cube of the scale factor. Therefore: \[ V_{\text{new}} = V_{\text{original}} \times (\text{scale factor})^3 = 336 \text{ m}^3 \times 3^3 \] \[ V_{\text{new}} = 336 \text{ m}^3 \times 27 = 9072 \text{ m}^3 \]
Thus, the new volume after the dilation is 9072 m³.
The correct answer is: 9,072 m³ or 9,072 m cubed.