Question
8 m
7 m
12 m
If the triangular prism were dilated using a scale factor of 3, what would be the new volume?
a. 27,216 m^3
b. 9,072 m^3
c. 112 m^3
d. 1,008 m^3
7 m
12 m
If the triangular prism were dilated using a scale factor of 3, what would be the new volume?
a. 27,216 m^3
b. 9,072 m^3
c. 112 m^3
d. 1,008 m^3
Answers
GPT-4o mini
To find the volume of a triangular prism, we first need to calculate the area of the triangular base and then use that to find the volume of the prism.
1. **Calculate the area of the triangular base:**
Given the triangle dimensions:
- Base = 8 m
- Height = 7 m
The area \( A \) of the triangle is calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 7 = 28 \, \text{m}^2
\]
2. **Calculate the volume of the prism before dilation:**
Given the length (or height) of the prism:
- Length = 12 m
The volume \( V \) of the prism is calculated using the formula:
\[
V = \text{Base Area} \times \text{Length} = 28 \, \text{m}^2 \times 12 \, \text{m} = 336 \, \text{m}^3
\]
3. **Apply the dilation:**
When a shape is dilated by a scale factor \( k \), the volume is scaled by \( k^3 \). In this case, the scale factor is 3, so:
\[
\text{New Volume} = \text{Original Volume} \times k^3 = 336 \, \text{m}^3 \times 3^3
\]
Calculating \( 3^3 \):
\[
3^3 = 27
\]
Now, calculate the new volume:
\[
\text{New Volume} = 336 \, \text{m}^3 \times 27 = 9072 \, \text{m}^3
\]
4. **Conclusion:**
The new volume after the dilation will be \( 9072 \, \text{m}^3 \).
The correct answer is **b. 9,072 m³**.
1. **Calculate the area of the triangular base:**
Given the triangle dimensions:
- Base = 8 m
- Height = 7 m
The area \( A \) of the triangle is calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 7 = 28 \, \text{m}^2
\]
2. **Calculate the volume of the prism before dilation:**
Given the length (or height) of the prism:
- Length = 12 m
The volume \( V \) of the prism is calculated using the formula:
\[
V = \text{Base Area} \times \text{Length} = 28 \, \text{m}^2 \times 12 \, \text{m} = 336 \, \text{m}^3
\]
3. **Apply the dilation:**
When a shape is dilated by a scale factor \( k \), the volume is scaled by \( k^3 \). In this case, the scale factor is 3, so:
\[
\text{New Volume} = \text{Original Volume} \times k^3 = 336 \, \text{m}^3 \times 3^3
\]
Calculating \( 3^3 \):
\[
3^3 = 27
\]
Now, calculate the new volume:
\[
\text{New Volume} = 336 \, \text{m}^3 \times 27 = 9072 \, \text{m}^3
\]
4. **Conclusion:**
The new volume after the dilation will be \( 9072 \, \text{m}^3 \).
The correct answer is **b. 9,072 m³**.