Question
8, 7 and 12. If the triangular prism were dilated using a scale factor of 3, what would be the new volume?
Answers
GPT-4o mini
To find the new volume of a triangular prism after dilation, we need to understand how dilation affects the volume.
Given the dimensions of the triangular prism:
- Base 1 (which could represent any of the triangular dimensions) = 8
- Base 2 = 7
- Height of the triangle = 12 (this is typically the height of the triangular face perpendicular to the base)
The volume \( V \) of a triangular prism can be calculated using the formula:
\[
V = \text{Area of the triangular base} \times \text{Height of the prism}
\]
To find the area of the triangular base, we use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Assuming the "base" of the triangle is 8 and the height is 12, the area \( A \) of the triangular base is:
\[
A = \frac{1}{2} \times 8 \times 12 = 48
\]
Now, if we assume the prism is rectangular with the triangular base as one of its faces and a height (length) equal to the third dimension (7, assuming that’s a constant distance), then the volume of the prism \( V \) is given by:
\[
V = \text{Area} \times \text{length} = 48 \times 7 = 336
\]
When the triangular prism is dilated by a scale factor of \( k \), the volume changes by a factor of \( k^3 \). Therefore, for a scale factor of 3:
\[
k = 3 \quad \Rightarrow \quad k^3 = 3^3 = 27
\]
The new volume \( V' \) after dilation is:
\[
V' = V \times k^3 = 336 \times 27
\]
Calculating \( 336 \times 27 \):
\[
336 \times 27 = 9072
\]
Therefore, the new volume of the triangular prism after dilation is \( \boxed{9072} \).
Given the dimensions of the triangular prism:
- Base 1 (which could represent any of the triangular dimensions) = 8
- Base 2 = 7
- Height of the triangle = 12 (this is typically the height of the triangular face perpendicular to the base)
The volume \( V \) of a triangular prism can be calculated using the formula:
\[
V = \text{Area of the triangular base} \times \text{Height of the prism}
\]
To find the area of the triangular base, we use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Assuming the "base" of the triangle is 8 and the height is 12, the area \( A \) of the triangular base is:
\[
A = \frac{1}{2} \times 8 \times 12 = 48
\]
Now, if we assume the prism is rectangular with the triangular base as one of its faces and a height (length) equal to the third dimension (7, assuming that’s a constant distance), then the volume of the prism \( V \) is given by:
\[
V = \text{Area} \times \text{length} = 48 \times 7 = 336
\]
When the triangular prism is dilated by a scale factor of \( k \), the volume changes by a factor of \( k^3 \). Therefore, for a scale factor of 3:
\[
k = 3 \quad \Rightarrow \quad k^3 = 3^3 = 27
\]
The new volume \( V' \) after dilation is:
\[
V' = V \times k^3 = 336 \times 27
\]
Calculating \( 336 \times 27 \):
\[
336 \times 27 = 9072
\]
Therefore, the new volume of the triangular prism after dilation is \( \boxed{9072} \).