Question
if the triangular prism (L = 12m, W = 7m, H = 8m) were dilated using a scale factor of 3, what would be the new volume
Answers
GPT-4o mini
To find the volume of a triangular prism, you can use the formula:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
For a triangular prism, the base area (where the base is a triangle) can be calculated using the formula for the area of a triangle:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height}_{\text{triangle}}
\]
Let's assume the dimensions of the triangle base are given by the length (L = 12 m) for the triangular base, and width (W = 7 m) corresponds to the height of the triangle.
1. **Calculate the base area of the triangle:**
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times L \times W = \frac{1}{2} \times 12 \, \text{m} \times 7 \, \text{m} = \frac{1}{2} \times 84 = 42 \, \text{m}^2
\]
2. **Now calculate the volume of the triangular prism:**
\[
\text{Volume} = \text{Base Area} \times H = 42 \, \text{m}^2 \times 8 \, \text{m} = 336 \, \text{m}^3
\]
3. **When the prism is dilated using a scale factor of 3, each dimension (length, width, and height) is multiplied by the scale factor. The volumes scale with the cube of the scale factor. Thus, the new volume \( V' \) is given by:**
\[
V' = \text{Volume} \times \text{scale factor}^3
\]
\[
V' = 336 \, \text{m}^3 \times 3^3
\]
\[
V' = 336 \, \text{m}^3 \times 27
\]
\[
V' = 9072 \, \text{m}^3
\]
Thus, the new volume of the dilated triangular prism is **9072 m³**.
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
For a triangular prism, the base area (where the base is a triangle) can be calculated using the formula for the area of a triangle:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height}_{\text{triangle}}
\]
Let's assume the dimensions of the triangle base are given by the length (L = 12 m) for the triangular base, and width (W = 7 m) corresponds to the height of the triangle.
1. **Calculate the base area of the triangle:**
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times L \times W = \frac{1}{2} \times 12 \, \text{m} \times 7 \, \text{m} = \frac{1}{2} \times 84 = 42 \, \text{m}^2
\]
2. **Now calculate the volume of the triangular prism:**
\[
\text{Volume} = \text{Base Area} \times H = 42 \, \text{m}^2 \times 8 \, \text{m} = 336 \, \text{m}^3
\]
3. **When the prism is dilated using a scale factor of 3, each dimension (length, width, and height) is multiplied by the scale factor. The volumes scale with the cube of the scale factor. Thus, the new volume \( V' \) is given by:**
\[
V' = \text{Volume} \times \text{scale factor}^3
\]
\[
V' = 336 \, \text{m}^3 \times 3^3
\]
\[
V' = 336 \, \text{m}^3 \times 27
\]
\[
V' = 9072 \, \text{m}^3
\]
Thus, the new volume of the dilated triangular prism is **9072 m³**.