To find the volume of the square-based pyramid, we first need to establish a relationship between the volume of the cube and the volume of the pyramid.
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Volume of the cube:
The volume \( V \) of a cube with side length \( s \) is given by:\[ V = s^3 \]
We know the volume of the cube is 330 cubic inches:
\[ s^3 = 330 \]
To find \( s \), compute the cube root:
\[ s = \sqrt[3]{330} \]
(We will compute the actual value of \( s \) later if needed.)
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Area of the base of the cube:
The area \( A \) of the base of the cube (which is a square) is:\[ A = s^2 \]
So, we can express the base area in terms of the volume:
\[ A = \sqrt[3]{330^2} \]
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Volume of the pyramid:
The volume \( V_p \) of a square-based pyramid is given by:\[ V_p = \frac{1}{3} \cdot \text{Base Area} \cdot \text{Height} \]
Since we know that the height of the pyramid is equal to the height of the cube and that the base area of the pyramid is the same as that of the cube (both equal \( s^2 \)), we can replace the base area and the height in the volume formula:
The height of the cube is \( s \), thus:
\[ V_p = \frac{1}{3} \cdot s^2 \cdot s = \frac{1}{3} s^3 \]
We already found \( s^3 = 330 \):
\[ V_p = \frac{1}{3} \cdot 330 = 110 \]
Therefore, the volume of the pyramid is \( 110 \) cubic inches.