To find the volume of a right triangular prism, we first need to calculate the volume of the original prism using the formula:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
The base of the triangular prism is a right triangle. The area \(A\) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we consider "width" to be the base and "height" to be 3 cm. Thus, the area of the base is:
\[ A = \frac{1}{2} \times 4 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 12 , \text{cm}^2 = 6 , \text{cm}^2 \]
Now, we can 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, the volume of the dilated prism can be found by applying the rule for the volume under dilation, which states that the new volume is the original volume multiplied by the cube of the scale factor:
\[ \text{Dilated Volume} = \text{Original Volume} \times (\text{scale factor})^3 \]
Here, the scale factor is 3, so:
\[ \text{Dilated Volume} = 42 , \text{cm}^3 \times 3^3 = 42 , \text{cm}^3 \times 27 = 1134 , \text{cm}^3 \]
Therefore, the volume of the dilated prism is:
\[ \boxed{1134 , \text{cm}^3} \]
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