To find the volume of a right triangular prism before and after dilation, we first need to calculate the volume of the original prism.
The formula for the volume of a prism is given by:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
For a right triangular prism, the base is a right triangle, and the area \(A\) of a right triangle can be calculated using:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base triangle has a length of 4 cm and a height of 3 cm. So, the area of the base triangle is:
\[ A = \frac{1}{2} \times 4 \times 3 = \frac{12}{2} = 6 \text{ cm}^2 \]
Next, we multiply the area of the base by the length of the prism (which is 7 cm) to find the volume:
\[ \text{Volume} = \text{Base Area} \times \text{Length} = 6 \times 7 = 42 \text{ cm}^3 \]
Now, to find the volume of the dilated prism, we use the fact that when a three-dimensional shape is dilated by a scale factor \(k\), the new volume is given by:
\[ \text{New Volume} = k^3 \times \text{Original Volume} \]
In this case, the scale factor \(k\) is 4. Therefore:
\[ \text{New Volume} = 4^3 \times 42 = 64 \times 42 \]
Calculating \(64 \times 42\):
\[ 64 \times 42 = 2688 \text{ cm}^3 \]
Thus, the volume of the dilated prism is:
\[ \boxed{2688 \text{ cm}^3} \]
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