To find the volume of the pyramid, we start with the information we have:
- The volume of the cube is given as 330 cubic inches.
- The height of both the cube and the pyramid are the same.
- The bases of both the cube and the pyramid have the same area.
First, we can find the side length of the cube. The volume \( V \) of a cube with side length \( s \) is given by the formula:
\[ V = s^3 \]
Given that \( V = 330 \) cubic inches, we can find the side length \( s \):
\[ s^3 = 330 \]
To compute \( s \), we take the cube root:
\[ s = \sqrt[3]{330} \]
Now we can also find the area of the base of the cube. The base of the cube is a square with area \( s^2 \):
\[ \text{Area of the base of the cube} = s^2 = (\sqrt[3]{330})^2 = 330^{2/3} \]
Now, since the base area of the pyramid is the same as that of the cube, the area of the base of the pyramid is also \( 330^{2/3} \).
Next, we need to determine the volume of the pyramid. The volume \( V \) of a pyramid with base area \( B \) and height \( h \) is given by:
\[ V = \frac{1}{3} B h \]
Since the pyramid has the same height as the cube, and we already have the area of the base \( B = 330^{2/3} \), we need to find \( h \). The height of the cube \( h \) is equal to the side length \( s \), which can be calculated as follows:
\[ h = \sqrt[3]{330} \]
Now substituting the values into the formula for the volume of the pyramid:
\[ V = \frac{1}{3} \times (330^{2/3}) \times (\sqrt[3]{330}) \]
We can simplify this expression. Notice that \( \sqrt[3]{330} = 330^{1/3} \).
Therefore,
\[ V = \frac{1}{3} \times 330^{2/3} \times 330^{1/3} \]
Using the property of exponents that states \( a^m \times a^n = a^{m+n} \):
\[ V = \frac{1}{3} \times 330^{(2/3 + 1/3)} = \frac{1}{3} \times 330^{1} = \frac{1}{3} \times 330 = 110 \]
Thus, the volume of the pyramid is:
\[ \boxed{110} \text{ cubic inches} \]