To find the volume of a right triangular prism, you can use the formula:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
In a right triangular prism, the base is a right triangle. The area of a right triangle can be calculated using:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
The dimensions provided are as follows:
- Width (base of the triangular base) = 4 cm
- Height (height of the triangular base) = 3 cm
- Length (height of the prism) = 7 cm
First, we calculate the area of the triangular base:
\[ \text{Area} = \frac{1}{2} \times 4 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 12 , \text{cm}^2 = 6 , \text{cm}^2 \]
Now, we calculate the volume of the prism:
\[ \text{Volume} = \text{Base Area} \times \text{Height} = 6 , \text{cm}^2 \times 7 , \text{cm} = 42 , \text{cm}^3 \]
Next, when the prism is dilated by a scale factor of 3, all dimensions (including the length, width, and height) are scaled by that factor.
The new dimensions after dilation will be:
- New Width = \(4 , \text{cm} \times 3 = 12 , \text{cm}\)
- New Length = \(7 , \text{cm} \times 3 = 21 , \text{cm}\)
- New Height = \(3 , \text{cm} \times 3 = 9 , \text{cm}\)
Now we need to recalculate the volume of the dilated prism.
- Calculate the area of the new triangular base:
\[ \text{New Area} = \frac{1}{2} \times 12 , \text{cm} \times 9 , \text{cm} = \frac{1}{2} \times 108 , \text{cm}^2 = 54 , \text{cm}^2 \]
- Calculate the new volume:
\[ \text{New Volume} = \text{New Base Area} \times \text{New Height} = 54 , \text{cm}^2 \times 21 , \text{cm} = 1134 , \text{cm}^3 \]
Thus, the volume of the dilated prism is:
\[ \boxed{1134 , \text{cm}^3} \]
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