To find the volume of the original right triangular prism, you can use the formula for the volume of a prism:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
However, we first need to determine the area of the base of the prism, which is a right triangle. The base (area of triangle) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Width} \times \text{Length} \]
Given:
- Width (W) = 4
- Length (L) = 7
Calculating the base area:
\[ \text{Base Area} = \frac{1}{2} \times 4 \times 7 = \frac{28}{2} = 14 \]
Now, using the height (H = 3) to find the volume of the original prism:
\[ \text{Volume} = \text{Base Area} \times \text{Height} = 14 \times 3 = 42 \]
Now, when the prism is dilated with a scale factor of 4, all dimensions (length, width, and height) are multiplied by the scale factor.
The volume of a dilated prism can be calculated using the formula for dilation of volume, which is the original volume multiplied by the cube of the scale factor:
\[ \text{Dilated Volume} = \text{Original Volume} \times \text{Scale Factor}^3 \]
Thus:
\[ \text{Dilated Volume} = 42 \times 4^3 = 42 \times 64 = 2688 \]
Therefore, the volume of the dilated prism would be 2688 cubic units.
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