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

1 m3
1 m cubed

30 m3
30 m cubed

6 m3
6 m cubed

3 m3

Answer 1

To find the volume of the pyramid, we can first recall the formulas for the volumes of a cube and a pyramid.

1. **Volume of the cube**:
The volume \( V \) of a cube with side length \( s \) is given by:
\[
V = s^3
\]

We know the volume of the cube is 9 cubic meters, so:
\[
s^3 = 9
\]
Thus, \( s = \sqrt[3]{9} \).

2. **Volume of the pyramid**:
The volume \( V \) of a square-based pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

In this case, since the cube and the pyramid share the same height and base area, we can calculate the base area. The base area of the pyramid is equal to the area of the base of the cube.

The base area of the cube is:
\[
\text{Base Area of Cube} = s^2 = \left(\sqrt[3]{9}\right)^2 = 9^{2/3}
\]
However, let’s just express it in cubic meters since we know the volume of the cube.

The height of the pyramid, which is the same as the side length of the cube, is \( \sqrt[3]{9} \).

Now substituting the base area and height into the pyramid's volume formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

The base area \( A \) is \( s^2 = (\sqrt[3]{9})^2 = 9^{2/3} \), and the height is \( s = \sqrt[3]{9}\).

Now we can express the volume of the pyramid:
\[
V = \frac{1}{3} \times 9^{2/3} \times 9^{1/3}
\]
Using properties of exponents:
\[
9^{2/3} \times 9^{1/3} = 9^{(2/3 + 1/3)} = 9^{3/3} = 9
\]
Thus,
\[
V = \frac{1}{3} \times 9 = 3 \text{ cubic meters}
\]

Therefore, the volume of the pyramid is \(\boxed{3 \text{ m}^3}\).