To find the volume of a right triangular prism, you first need to calculate the area of the triangular base and then multiply that area by the height of the prism.
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Calculate the area of the triangular base: The base of the triangular prism is a right triangle. The area \( A \) of a right triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Here, we can take the two legs of the triangle as the base and height:
- Base \( b = 4 \) cm
- Height \( h = 3 \) cm (using the same height as the prism's height is common).
So the area is: \[ A = \frac{1}{2} \times 4 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 12 , \text{cm}^2 = 6 , \text{cm}^2 \]
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Calculate the volume of the prism: The volume \( V \) of the prism can be computed by multiplying the area of the base by the length: \[ V = \text{base area} \times \text{length} \] Given the length \( l = 7 \) cm: \[ V = 6 , \text{cm}^2 \times 7 , \text{cm} = 42 , \text{cm}^3 \]
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Dilating the prism: When a prism is dilated with a scale factor \( k \), the volume of the new prism is given by: \[ V' = k^3 \times V \] In this case, \( k = 3 \): \[ V' = 3^3 \times 42 , \text{cm}^3 = 27 \times 42 , \text{cm}^3 \]
Now, calculate \( 27 \times 42 \): \[ V' = 27 \times 42 = 1134 , \text{cm}^3 \]
Thus, the volume of the dilated prism is 1134 cm³.
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