To solve for the volume of the pyramid, we first need to understand the formulas for the volume of both a cube and a pyramid.
-
Volume of the cube: The formula for the volume of a cube is given by: \[ V_{\text{cube}} = s^3 \] where \(s\) is the length of a side of the cube.
Given that \(V_{\text{cube}} = 330\) cubic inches, we have: \[ s^3 = 330 \]
-
Volume of the pyramid: The formula for the volume of a square-based pyramid is given by: \[ V_{\text{pyramid}} = \frac{1}{3} B h \] where \(B\) is the area of the base and \(h\) is the height.
Since it's stated that the pyramid has the same height and the same base area as the base of the cube, we proceed as follows:
-
Calculate the side length of the cube: To find \(s\), we take the cube root: \[ s = \sqrt[3]{330} \]
Now, using a calculator: \[ s \approx 6.879 \]
-
Base area of the cube: The base area \(B\) of the cube is: \[ B = s^2 \approx (6.879)^2 \approx 47.3 \text{ square inches} \]
-
Height of the pyramid: The height \(h\) of the cube is equal to its side length and hence: \[ h = s \approx 6.879 \text{ inches} \]
-
Volume of the pyramid: Now, substituting \(B\) and \(h\) into the volume formula for the pyramid: \[ V_{\text{pyramid}} = \frac{1}{3} B h \] \[ V_{\text{pyramid}} \approx \frac{1}{3} \times 47.3 \times 6.879 \]
Calculating this: \[ V_{\text{pyramid}} \approx \frac{1}{3} \times 324.16 \approx 108.72 \text{ cubic inches} \]
Thus, the volume of the pyramid is approximately 108.72 cubic inches.