To find the volumes of the triangular prism and pyramid that both use the same triangular base, follow these steps:
1. Calculate the area of the triangular base.
The base of the triangle is an equilateral triangle with a side length of 6 cm. The formula for the area \( A \) of an equilateral triangle is:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Where \( s \) is the side length.
Plugging in the given side length:
\[ A = \frac{\sqrt{3}}{4} (6)^2 = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3} \text{ cm}^2 \]
2. Calculate the volume of the triangular prism.
The formula for the volume \( V \) of a prism is:
\[ V = \text{Base Area} \times \text{Height} \]
The height of the prism is given as 9 cm. Therefore:
\[ V_{\text{prism}} = 9\sqrt{3} \times 9 = 81\sqrt{3} \text{ cm}^3 \]
Calculating \( 81\sqrt{3} \):
Since \( \sqrt{3} \) is approximately 1.732:
\[ V_{\text{prism}} \approx 81 \times 1.732 \approx 140.232 \text{ cm}^3 \]
3. Calculate the volume of the triangular pyramid.
The formula for the volume \( V \) of a pyramid is:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Using the same base area \( 9\sqrt{3} \) cm² and a height of 9 cm:
\[ V_{\text{pyramid}} = \frac{1}{3} \times 9\sqrt{3} \times 9 = \frac{81\sqrt{3}}{3} = 27\sqrt{3} \text{ cm}^3 \]
Calculating \( 27\sqrt{3} \):
\[ V_{\text{pyramid}} \approx 27 \times 1.732 \approx 46.704 \text{ cm}^3 \]
4. Rounding to the nearest whole numbers:
Finally, rounding \( 140.232 \) and \( 46.704 \) gives us:
- Volume of the prism: approximately 140 cm³
- Volume of the pyramid: approximately 47 cm³
5. Select the closest option:
Comparing the calculated volumes with the options given:
- Prism: 216 cm³; Pyramid: 72 cm³
- Prism: 72 cm³; Pyramid: 216 cm³
- Prism: 108 cm³; Pyramid: 36 cm³
- Prism: 36 cm³; Pyramid: 108 cm³
None of the options exactly match, but it appears the information given might not be aligned with my calculations. If we strictly regard only the calculation process without numerical errors:
- Prism: 216 cm³ indicates a wrong base area interpretation with revised height considerations.
- Pyramid: 72 cm³ might be re-evaluated in respect to the pyramid's output derived geometrically.
Finalize with detailed review or possibly reconciling the specifications asked within problem scope.