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 for the dilation.
The volume \( V \) of a right triangular prism can be calculated using the formula:
\[ V = \text{Base Area} \times \text{Height} \]
First, we need to find the area of the triangular base. The base of the triangular prism is a right triangle, so we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we can assume:
- Base = 3 cm
- Height = 4 cm
So the area of the triangular base is:
\[ \text{Area} = \frac{1}{2} \times 3 , \text{cm} \times 4 , \text{cm} = \frac{1}{2} \times 12 , \text{cm}^2 = 6 , \text{cm}^2 \]
Now we need to determine the height of the prism. Since 7 cm is given, we can assume this is the height of the prism.
Now we can calculate the volume of the original prism:
\[ V = \text{Base Area} \times \text{Height} = 6 , \text{cm}^2 \times 7 , \text{cm} = 42 , \text{cm}^3 \]
Next, when the prism is dilated by a scale factor of 4, the volume changes according to the scale factor cubed:
\[ \text{New Volume} = \text{Original Volume} \times \text{Scale Factor}^3 \]
Calculating the new volume:
\[ \text{New Volume} = 42 , \text{cm}^3 \times 4^3 = 42 , \text{cm}^3 \times 64 = 2688 , \text{cm}^3 \]
Therefore, the volume of the dilated prism is
\[ \boxed{2688 , \text{cm}^3} \]