To find the volume of the dilated right triangular prism, we first need to calculate the volume of the original prism and then apply the scale factor to determine the volume of the dilated prism.
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Calculate the volume of the original prism: The volume \( V \) of a triangular prism can be calculated using the formula: \[ V = \text{Base Area} \times \text{Height} \]
Given that the base of the prism is a right triangle with legs of lengths 4 and 7, the area of the triangle can be calculated as: \[ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 7 = \frac{28}{2} = 14 \]
The height of the prism is given as 3. Therefore, the volume of the original prism is: \[ V = 14 \times 3 = 42 \]
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Determine the scale factor: The prism is dilated using a scale factor of 4. When an object is dilated by a scale factor, the volume changes by the cube of that scale factor. Therefore, the new volume \( V' \) of the dilated prism can be calculated as: \[ V' = V \times k^3 \] where \( k \) is the scale factor.
Here, \( k = 4 \): \[ V' = 42 \times 4^3 = 42 \times 64 = 2688 \]
Thus, the volume of the dilated prism is \( \boxed{2688} \).
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