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

1m3
1 m cubed

30m3
30m cubed

6m3

Answer 1

To find the volume of the square-based pyramid, we start with the information given:

1. The volume of the cube is 9 cubic meters.
2. The height of the pyramid is equal to the side length of the cube.
3. The base area of the pyramid is equal to the area of the base of the cube.

First, since the volume of a cube is calculated as:
\[
V = s^3
\]
where \( s \) is the side length, we have:
\[
s^3 = 9 \quad \Rightarrow \quad s = \sqrt[3]{9} \approx 2.08 \, \text{m}
\]

Next, we can calculate the base area of the cube. The base area (which is a square) is:
\[
\text{Base Area} = s^2 = (\sqrt[3]{9})^2 = 9^{2/3} \, \text{m}^2
\]

Now, we can find the volume of the pyramid. The volume \( V \) of a square-based pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

Given that:
- Base Area = \( 9^{2/3} \)
- Height = \( s = \sqrt[3]{9} \)

We can substitute these values into the formula for the volume of the pyramid:
\[
V = \frac{1}{3} \times 9^{2/3} \times \sqrt[3]{9}
\]

Now, we simplify:
\[
= \frac{1}{3} \times 9^{2/3} \times 9^{1/3} = \frac{1}{3} \times 9^{(2/3 + 1/3)} = \frac{1}{3} \times 9^{1} = \frac{9}{3} = 3
\]

Thus, the volume of the pyramid is:
\[
\boxed{3 \, \text{m}^3}
\]