To find the volume of the square-based pyramid given that it has the same height and base area as a cube with a volume of 54 cubic units, we can follow these steps:
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Calculate the side length of the cube: The volume \( V \) of the cube is given by the formula: \[ V = s^3 \] where \( s \) is the side length. Given that the volume is 54 cubic units, we find \( s \): \[ s^3 = 54 \implies s = \sqrt[3]{54} = 3 \sqrt[3]{2} \quad (\text{approximately } 3.78) \] We will use exact values for further calculations.
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Calculate the base area of the cube: The base area \( A \) of the cube (which is a square) is: \[ A = s^2 \] Substituting \( s = \sqrt[3]{54} \): \[ A = (s)^2 = (s^3)^{2/3} = 54^{2/3} \]
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Calculate the height of the cube: The height \( h \) of the cube is equal to its side length \( s \): \[ h = s = \sqrt[3]{54} \]
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Calculate the volume of the pyramid: The volume \( V_p \) of a square-based pyramid is given by the formula: \[ V_p = \frac{1}{3} A h \] We plug in the values for base area \( A \) and height \( h \): \[ V_p = \frac{1}{3} (54^{2/3})(54^{1/3}) = \frac{1}{3} (54^{2/3 + 1/3}) = \frac{1}{3} (54^{3/3}) = \frac{1}{3} (54) = 18 \text{ cubic units} \]
So, the volume of the pyramid is 18 cubic units.
The correct answer is: 18 cubic units.
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