To solve the problem, let's first recall the formulas for the volume of a cube and a square-based pyramid.
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Volume of a Cube (V_cube): The volume of a cube is given by: \[ V_{\text{cube}} = s^3 \] where \(s\) is the length of a side of the cube.
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Volume of a Pyramid (V_pyramid): The volume of a square-based pyramid is given by: \[ V_{\text{pyramid}} = \frac{1}{3} \times B \times h \] where \(B\) is the area of the base and \(h\) is the height of the pyramid.
Since the volume of the cube is given as 54 cubic units, we can find the side length \(s\): \[ s^3 = 54 \implies s = \sqrt[3]{54} = 3 \sqrt[3]{2} \]
Given that the pyramid and the cube have the same base area and height, we will first find the base area of the cube:
- The base area of the cube is: \[ B_{\text{cube}} = s^2 = (3 \sqrt[3]{2})^2 = 9 \cdot \sqrt[3]{4} \]
Since the pyramid has the same base area and height as the cube, we can denote:
- Base area \(B\) of the pyramid: \[ B = 9 \cdot \sqrt[3]{4} \]
- Height \(h\) of both the cube and pyramid (which is also \(s\), because the height of the cube is equal to its side): \[ h = s = 3 \sqrt[3]{2} \]
Now we can calculate the volume of the pyramid: \[ V_{\text{pyramid}} = \frac{1}{3} \times B \times h = \frac{1}{3} \times (9 \cdot \sqrt[3]{4}) \times (3 \cdot \sqrt[3]{2}) \] \[ = \frac{1}{3} \times 27 \cdot \sqrt[3]{8} = 9 \cdot 2 = 18 \]
Thus, the volume of the pyramid is 18 cubic units.
Final answer: 18 cubic units.
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