To find the volume of a triangular prism and a triangular pyramid, we can use the following formulas:
-
Volume of a triangular prism: \[ V_{\text{prism}} = \text{Base Area} \times \text{Height} \]
-
Volume of a triangular pyramid: \[ V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Given that both shapes have a height of 9 cm, we can determine the volumes based on the area of the triangular base.
Let’s assume the base area of the triangular base is \( A \).
-
Volume of the prism: \[ V_{\text{prism}} = A \times 9 \]
-
Volume of the pyramid: \[ V_{\text{pyramid}} = \frac{1}{3} \times A \times 9 = 3A \]
From the relationship between the volumes, we see that the volume of the prism is three times that of the pyramid: \[ V_{\text{prism}} = 3 \times V_{\text{pyramid}} \]
Since we know that the volumes can be directly compared once we know the area \( A \), we can summarize the conditions based on this ratio.
Now, let's analyze the provided options:
-
Option 1: prism: 216 cm³; pyramid: 72 cm³
- \( V_{\text{prism}} = 3 \times V_{\text{pyramid}} \) → \( 216 = 3 \times 72 \) (True)
-
Option 2: prism: 36 cm³; pyramid: 108 cm³
- \( V_{\text{prism}} = 3 \times V_{\text{pyramid}} \) → \( 36 = 3 \times 108 \) (False)
-
Option 3: prism: 72 cm³; pyramid: 216 cm³
- \( V_{\text{prism}} = 3 \times V_{\text{pyramid}} \) → \( 72 = 3 \times 216 \) (False)
-
Option 4: prism: 108 cm³; pyramid: 36 cm³
- \( V_{\text{prism}} = 3 \times V_{\text{pyramid}} \) → \( 108 = 3 \times 36 \) (True)
Based on this analysis, the correct responses are:
- Response: prism: 216 cm³; pyramid: 72 cm³
- Response: prism: 108 cm³; pyramid: 36 cm³
Both options 1 and 4 are mathematically valid based on the information given, but option 1 (prism: 216 cm³; pyramid: 72 cm³) seems to be the most logically appropriate representation since it aligns with commonly encountered values in educational contexts.
Final conclusion:
- Prism: 216 cm³
- Pyramid: 72 cm³