We are given that the volume of the cube is 450 cubic feet, and we need to find the volume of a square-based pyramid that has the same base area and height as the cube.
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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 side length of the cube.
Given that the volume of the cube is 450 cubic feet, we can write: \[ s^3 = 450 \] To find the side length \( s \), we take the cube root: \[ s = \sqrt[3]{450} \]
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Base Area of the Cube: The base area \( A \) of the cube is: \[ A = s^2 \] So substituting \( s = \sqrt[3]{450} \): \[ A = \left(\sqrt[3]{450}\right)^2 = \sqrt[3]{450^2} \]
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Height of the Cube: The height of the cube is simply equal to its side length \( s \): \[ h = s = \sqrt[3]{450} \]
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Volume of the Pyramid: The volume \( V_p \) of a square-based pyramid is given by: \[ V_p = \frac{1}{3} A h \] where \( A \) is the base area and \( h \) is the height.
Using the base area \( A = s^2 \) and height \( h = s \), we substitute these into the volume formula: \[ V_p = \frac{1}{3} (s^2)(s) = \frac{1}{3} s^3 \]
Since \( s^3 = 450 \), we have: \[ V_p = \frac{1}{3} \times 450 = 150 \]
Thus, the volume of the pyramid is \(\boxed{150}\) cubic feet.
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