To find the volume of the pyramid, we first need to determine the dimensions of the cube and then use those to find the pyramid's volume.
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Volume of the cube: The volume \( V \) of a cube is given by the formula: \[ V = s^3 \] where \( s \) is the length of a side of the cube.
Given that the volume of the cube is 9 cubic meters: \[ s^3 = 9 \] To find \( s \), we take the cube root: \[ s = \sqrt[3]{9} \approx 2.08 \text{ meters} \]
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Height and base area of the pyramid: Mei Li notes that the pyramid has the same height and base area as the cube. Thus:
- The height \( h \) of the pyramid is \( s \) (since the height of the cube is also its side length).
- The base area \( A \) of the cube is \( s^2 \).
We can calculate the base area: \[ A = s^2 = (\sqrt[3]{9})^2 = 9^{2/3} \approx 6.76 \text{ square meters} \]
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Volume of the pyramid: The volume \( V \) of a pyramid is given by the formula: \[ V = \frac{1}{3} A h \] Substituting \( A \) and \( h \): \[ A = 9^{2/3} \] and \[ h = \sqrt[3]{9} \] Therefore, the volume becomes: \[ V = \frac{1}{3} (9^{2/3}) (9^{1/3}) = \frac{1}{3} \times 9^{(2/3 + 1/3)} = \frac{1}{3} \times 9^{1} = \frac{9}{3} = 3 \text{ cubic meters} \]
Thus, the volume of the pyramid is 3 m³.
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