To find the new volume after dilating a triangular prism with a scale factor, we first need to calculate the original volume and then apply the effect of the dilation.
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Calculate the volume of the original triangular prism: The formula for the volume \(V\) of a triangular prism is given by:
\[ V = \text{Base Area} \times \text{Height} \]
The base of the triangular prism is a triangle. We don't have the dimensions of the triangle directly, but we can use the given width and height. Assuming that width (7) represents one side of the triangle and that the height of the triangle is also 8. We will calculate the area of the triangle first.
The area \(A\) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Here, Base = 7 and Height = 8:
\[ A = \frac{1}{2} \times 7 \times 8 = \frac{56}{2} = 28 \text{ m}^2 \]
Now, the volume of the triangular prism (length = 12):
\[ V = A \times \text{Length} = 28 \text{ m}^2 \times 12 \text{ m} = 336 \text{ m}^3 \]
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Apply the scale factor: When a three-dimensional object is dilated by a scale factor \(k\), the volume is multiplied by \(k^3\). Here, the scale factor is 3.
\[ \text{New Volume} = \text{Original Volume} \times k^3 = 336 \text{ m}^3 \times 3^3 \]
Calculate \(3^3\):
\[ 3^3 = 27 \]
Therefore,
\[ \text{New Volume} = 336 \text{ m}^3 \times 27 = 9072 \text{ m}^3 \]
So, the new volume after dilation with a scale factor of 3 is:
9,072 m³
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