Question

Latoria has a cube and a square-based pyramid. The two figures have the same height and their bases have the same area. Latoria knows that the volume of the cube is 330 cubic inches. What is the volume of the pyramid?(1 point)The volume of the pyramid is  in.3.

Answer 1

To find the volume of the square-based pyramid, we can use the relationship between the volumes of the cube and pyramid and the fact that both have the same height and base area.

1. **Volume of the Cube**:
The volume \( V \) of a cube is given by the formula:
\[
V = s^3
\]
where \( s \) is the length of a side of the cube. We know from the problem that the volume of the cube is \( 330 \) cubic inches:
\[
s^3 = 330
\]

2. **Finding Side Length of the Cube**:
We can find \( s \) by taking the cube root of \( 330 \):
\[
s = \sqrt[3]{330} \approx 6.885
\]

3. **Volume of the Pyramid**:
The volume \( V \) of a square-based pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The base area of the pyramid is the same as the base area of the cube, which is \( s^2 \). Thus, we calculate the base area:
\[
\text{Base Area} = s^2 = (\sqrt[3]{330})^2 = \sqrt[3]{330^2}
\]

4. **Height**:
The height of the pyramid is also \( s \) (the side of the cube).

5. **Combining the Formulas for Volume**:
With base area \( s^2 \) and height \( s \), the volume of the pyramid becomes:
\[
V_{\text{pyramid}} = \frac{1}{3} \times s^2 \times s = \frac{1}{3} \times s^3 = \frac{1}{3} \times 330 = 110
\]
Thus, the volume of the pyramid is \( \boxed{110} \) cubic inches.