To find the volume of the dilated right triangular prism, we first need to determine the volume of the original prism and then apply the dilation.
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Finding the area of the triangular base: The triangle formed by the sides 3 cm, 4 cm, and the height corresponding to the base of 4 cm. We can use the formula for the area of a right triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Here, we can consider the base as 4 cm and the height as 3 cm: \[ \text{Area} = \frac{1}{2} \times 4 \times 3 = \frac{12}{2} = 6 \text{ cm}^2 \]
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Finding the volume of the original prism: The volume \( V \) of a prism is given by the formula: \[ V = \text{Base Area} \times \text{Height} \] In this case, the height of the prism is 7 cm: \[ V = 6 \times 7 = 42 \text{ cm}^3 \]
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Applying the scale factor: When a prism is dilated with a scale factor of \( k \), its volume changes by a factor of \( k^3 \). In this case, \( k = 3 \): \[ \text{New Volume} = \text{Original Volume} \times k^3 = 42 \times 3^3 = 42 \times 27 = 1134 \text{ cm}^3 \]
Thus, the volume of the dilated triangular prism would be 1134 cm³.
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