To solve the problem, we need to use the formulas for the volumes of the cube and the square-based pyramid.
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Volume of the cube: The volume \( V \) of a cube is calculated using the formula: \[ V = s^3 \] where \( s \) is the length of one side of the cube. Given that the volume of the cube is 54 cubic units, we can find the side length: \[ s^3 = 54 \implies s = \sqrt[3]{54} \approx 3.78 , \text{units} \]
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Volume of the pyramid: The volume \( V \) of a square-based pyramid is given by the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Given that the base area of the pyramid is the same as that of the cube, we can find the base area. The base area of the cube (which is square) is: \[ \text{Base Area} = s^2 = (\sqrt[3]{54})^2 = 54^{2/3} \]
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Height: The height of the cube is equal to its side length \( s \), which we already approximated to be \( \sqrt[3]{54} \).
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Calculating base area: \[ \text{Base Area} = s^2 = (\sqrt[3]{54})^2 = \sqrt[3]{54^2} \] This can also be simplified to: \[ \text{Base Area} = 54^{2/3} \approx 14.4 , \text{square units} \]
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Substituting into the volume formula for the pyramid: Since both height (which is like the side of the cube) and base area (same as the area of the cube's base) are equal: \[ \text{Height} = \sqrt[3]{54} \] \[ \text{Base Area} = 54^{2/3} \]
The volume of the pyramid can now be computed: \[ V_{\text{pyramid}} = \frac{1}{3} \times (54^{2/3}) \times (\sqrt[3]{54}) = \frac{1}{3} \times \sqrt[3]{54^3} = \frac{1}{3} \times 54 = 18 , \text{cubic units} \]
Thus, the volume of the pyramid is: \[ \boxed{18} , \text{cubic units} \]