To determine the new volume of a triangular prism after dilation, you need to know the volume of the original prism and how dilation affects the volume.
First, calculate the volume of the original triangular prism. The formula for the volume of a triangular prism is:
\[ V = \text{Base Area} \times \text{Height} \]
The base area of a triangular prism can be calculated using the formula for the area of a triangle. However, you have given values for length (L), width (W), and height (H). Let's assume L is the base length, W is the width (or the height of the triangle), and H is the height of the prism itself. If the triangle is right-angled, the area of the base (triangle) can be calculated as:
\[ \text{Area} = \frac{1}{2} \times L \times W \]
Plugging in the values:
\[ \text{Area} = \frac{1}{2} \times 12 , \text{m} \times 7 , \text{m} = \frac{1}{2} \times 84 , \text{m}^2 = 42 , \text{m}^2 \]
Now, calculate the original volume:
\[ V = \text{Base Area} \times \text{Height} = 42 , \text{m}^2 \times 8 , \text{m} = 336 , \text{m}^3 \]
Next, when a three-dimensional shape is dilated by a scale factor \(k\), the volume changes by a factor of \(k^3\). Here, the scale factor is 3, so we calculate:
\[ \text{New Volume} = \text{Original Volume} \times k^3 = 336 , \text{m}^3 \times 3^3 = 336 , \text{m}^3 \times 27 \]
Now, calculate that:
\[ 336 , \text{m}^3 \times 27 = 9072 , \text{m}^3 \]
Thus, the new volume of the dilated triangular prism is 9072 m³.
The correct response from your options would be:
9,072 m cubed.
1 answer