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?(1 point)
Responses

3 m3
3 m cubed

30m3
30m cubed

6m3
6m cubed

1m3

Answer 1

To find the volume of the pyramid, we first need to determine the dimensions of the cube and then use those to find the pyramid's volume.

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.

Given that the volume of the cube is 9 cubic meters:
\[
s^3 = 9
\]
To find \( s \), we take the cube root:
\[
s = \sqrt[3]{9} \approx 2.08 \text{ meters}
\]

2. **Height and base area of the pyramid**: Mei Li notes that the pyramid has the same height and base area as the cube. Thus:
- The height \( h \) of the pyramid is \( s \) (since the height of the cube is also its side length).
- The base area \( A \) of the cube is \( s^2 \).

We can calculate the base area:
\[
A = s^2 = (\sqrt[3]{9})^2 = 9^{2/3} \approx 6.76 \text{ square meters}
\]

3. **Volume of the pyramid**: The volume \( V \) of a pyramid is given by the formula:
\[
V = \frac{1}{3} A h
\]
Substituting \( A \) and \( h \):
\[
A = 9^{2/3}
\]
and
\[
h = \sqrt[3]{9}
\]
Therefore, the volume becomes:
\[
V = \frac{1}{3} (9^{2/3}) (9^{1/3}) = \frac{1}{3} \times 9^{(2/3 + 1/3)} = \frac{1}{3} \times 9^{1} = \frac{9}{3} = 3 \text{ cubic meters}
\]

Thus, the volume of the pyramid is **3 m³**.