To find the volume of the pyramid given that the cube and pyramid have identical heights and the same base area, we can use the following formulas:
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Volume of a cube: The volume \( V \) is given by the formula: \[ V = s^3 \] where \( s \) is the length of a side of the cube.
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Volume of a pyramid: The volume \( V \) of a pyramid is given by the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Given that the volume of the cube is 450 cubic feet, we can set up the equation: \[ s^3 = 450 \] From this, we can find the side length \( s \): \[ s = \sqrt[3]{450} \]
Next, since the pyramid has the same base area as the cube, and both have the same height \( h \), we need to find the base area of the cube. The base area of the cube, which is a square, is: \[ \text{Base Area} = s^2 \]
Now we need to establish the relation between the volumes.
The volume of the pyramid can be expressed as: \[ V_{\text{pyramid}} = \frac{1}{3} \times s^2 \times h \]
We need to find the height \( h \) of the cube. Since the volume of the cube is also given by: \[ 450 = s^2 \times h \]
We already have expressed \( h \) in terms of \( s \): \[ h = \frac{450}{s^2} \]
Substituting \( h \) back into the volume of the pyramid, we get: \[ V_{\text{pyramid}} = \frac{1}{3} \times s^2 \times \frac{450}{s^2} = \frac{1}{3} \times 450 = 150 \]
Thus, the volume of the pyramid is: \[ \boxed{150 \text{ ft}^3} \]