To find the volume of the square-based pyramid, we first need to determine the side length of the cube and subsequently use the information given to find the properties of the pyramid.
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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 330 cubic inches, we can set up the equation: \[ s^3 = 330 \]
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To find \( s \), we take the cube root of both sides: \[ s = \sqrt[3]{330} \approx 6.879 \text{ inches} \]
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Since the pyramid has the same height as the cube, the height \( h \) of the pyramid is also \( s \): \[ h = s = \sqrt[3]{330} \approx 6.879 \text{ inches} \]
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The base of the cube is a square with an area \( A \) equal to \( s^2 \): \[ A = s^2 = (\sqrt[3]{330})^2 \approx 47.271 \text{ square inches} \]
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The volume \( V_p \) of a square-based pyramid is given by the formula: \[ V_p = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Plugging in the values we have: \[ V_p = \frac{1}{3} \times A \times h = \frac{1}{3} \times s^2 \times s = \frac{1}{3} \times s^3 = \frac{1}{3} \times 330 = 110 \text{ cubic inches} \]
Thus, the volume of the pyramid is \( \boxed{110} \) cubic inches.