Question

Mei Li measures a cube and a square-based pyramid and finds that the pyramid has the same height and base area as the cube. She also calculates that the volume of the cube is 9 cubic meters. What is the volume of the pyramid?

Answer 1

To find the volume of the pyramid, we first need to consider the given information about the cube.

1. **Calculate the side length of the cube**:
Given that the volume of the cube \( V \) is given by the formula:
\[
V = s^3
\]
where \( s \) is the side length of the cube. We know from the problem that:
\[
V = 9 \text{ cubic meters}
\]
Therefore, we can solve for \( s \):
\[
s^3 = 9
\]
Taking the cube root of both sides, we find:
\[
s = \sqrt[3]{9} \approx 2.08 \text{ meters}
\]

2. **Identify the height and base area of the pyramid**:
The pyramid has the same height and base area as the cube. Hence:
- The height \( h \) of the pyramid is equal to the side length of the cube, which is \( \sqrt[3]{9} \).
- The base area \( A \) of the pyramid is equal to the area of the base of the cube, since the base of a cube (which is a square) has area:
\[
A = s^2 = (\sqrt[3]{9})^2 = 9^{2/3} \text{ square meters}
\]

3. **Calculate the volume of the pyramid**:
The volume \( V \) of a pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the base area and height we found:
\[
V = \frac{1}{3} \times 9^{2/3} \times \sqrt[3]{9}
\]

We can simplify this expression step by step. Note that \( \sqrt[3]{9} = 9^{1/3} \). Thus, we can rewrite:
\[
V = \frac{1}{3} \times 9^{2/3} \times 9^{1/3} = \frac{1}{3} \times 9^{(2/3 + 1/3)} = \frac{1}{3} \times 9^{1} = \frac{1}{3} \times 9 = 3 \text{ cubic meters}
\]

4. **Final Answer**:
Therefore, the volume of the pyramid is:
\[
\boxed{3} \text{ cubic meters}
\]