Question
calculate and compare the volume of the two triangular prism figures. Figure 1 H=27.5 B=27.5 L=55 and figure 2 H=5.5 B=5.5 L=11 in the image. choose which statement is true.
the two figures are similar because the volumes have a ratio of 125 : 1
the two figures are not similar because there is no scale factor stated for the volume
the two figures are not similar because the volumes are not the same
the two fiures are similar because the volumes have a ratio of 5 : 1
the two figures are similar because the volumes have a ratio of 125 : 1
the two figures are not similar because there is no scale factor stated for the volume
the two figures are not similar because the volumes are not the same
the two fiures are similar because the volumes have a ratio of 5 : 1
Answers
GPT-4o mini
To determine if the two triangular prisms are similar and to evaluate the volume ratio, we need to calculate the volume of each prism using the formula for the volume of a triangular prism:
\[
\text{Volume} = \text{Base Area} \times \text{Length}
\]
For a triangular prism, the base area (A) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
### Figure 1:
- Height (H1) = 27.5
- Base (B1) = 27.5
- Length (L1) = 55
First, calculate the area of the base:
\[
A_1 = \frac{1}{2} \times B_1 \times H_1 = \frac{1}{2} \times 27.5 \times 27.5 = \frac{1}{2} \times 756.25 = 378.125
\]
Now, calculate the volume of Figure 1:
\[
V_1 = A_1 \times L_1 = 378.125 \times 55 = 20796.875
\]
### Figure 2:
- Height (H2) = 5.5
- Base (B2) = 5.5
- Length (L2) = 11
Now, calculate the area of the base for Figure 2:
\[
A_2 = \frac{1}{2} \times B_2 \times H_2 = \frac{1}{2} \times 5.5 \times 5.5 = \frac{1}{2} \times 30.25 = 15.125
\]
Now, calculate the volume of Figure 2:
\[
V_2 = A_2 \times L_2 = 15.125 \times 11 = 166.375
\]
### Compare the Volumes:
Now, we can determine the ratio of the volumes:
\[
\text{Volume Ratio} = \frac{V_1}{V_2} = \frac{20796.875}{166.375} \approx 125.0
\]
Thus, the volumes of the two figures are in a ratio of approximately 125:1.
### Conclusion:
Now let's evaluate the given statements:
1. **The two figures are similar because the volumes have a ratio of 125 : 1** - **True**.
2. **The two figures are not similar because there is no scale factor stated for the volume** - False.
3. **The two figures are not similar because the volumes are not the same** - False.
4. **The two figures are similar because the volumes have a ratio of 5 : 1** - False.
Therefore, the statement that is true is:
**The two figures are similar because the volumes have a ratio of 125 : 1.**
\[
\text{Volume} = \text{Base Area} \times \text{Length}
\]
For a triangular prism, the base area (A) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
### Figure 1:
- Height (H1) = 27.5
- Base (B1) = 27.5
- Length (L1) = 55
First, calculate the area of the base:
\[
A_1 = \frac{1}{2} \times B_1 \times H_1 = \frac{1}{2} \times 27.5 \times 27.5 = \frac{1}{2} \times 756.25 = 378.125
\]
Now, calculate the volume of Figure 1:
\[
V_1 = A_1 \times L_1 = 378.125 \times 55 = 20796.875
\]
### Figure 2:
- Height (H2) = 5.5
- Base (B2) = 5.5
- Length (L2) = 11
Now, calculate the area of the base for Figure 2:
\[
A_2 = \frac{1}{2} \times B_2 \times H_2 = \frac{1}{2} \times 5.5 \times 5.5 = \frac{1}{2} \times 30.25 = 15.125
\]
Now, calculate the volume of Figure 2:
\[
V_2 = A_2 \times L_2 = 15.125 \times 11 = 166.375
\]
### Compare the Volumes:
Now, we can determine the ratio of the volumes:
\[
\text{Volume Ratio} = \frac{V_1}{V_2} = \frac{20796.875}{166.375} \approx 125.0
\]
Thus, the volumes of the two figures are in a ratio of approximately 125:1.
### Conclusion:
Now let's evaluate the given statements:
1. **The two figures are similar because the volumes have a ratio of 125 : 1** - **True**.
2. **The two figures are not similar because there is no scale factor stated for the volume** - False.
3. **The two figures are not similar because the volumes are not the same** - False.
4. **The two figures are similar because the volumes have a ratio of 5 : 1** - False.
Therefore, the statement that is true is:
**The two figures are similar because the volumes have a ratio of 125 : 1.**