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
6 m3
6 m cubed
3 m3
3 m cubed
30 m3
30 m cubed
1 m3
Responses
6 m3
6 m cubed
3 m3
3 m cubed
30 m3
30 m cubed
1 m3
Answers
GPT-4o mini
To find the volume of the pyramid, we first need to determine the dimensions of the cube and the pyramid based on the information given.
1. **Volume of the cube**: The volume of a cube is calculated using the formula:
\[
V_{\text{cube}} = s^3
\]
where \( s \) is the side length of the cube. Given that the volume of the cube is 9 cubic meters, we can set up the equation:
\[
s^3 = 9
\]
To find \( s \), take the cube root:
\[
s = \sqrt[3]{9}
\]
2. **Height and base area of the pyramid**: The pyramid has the same height and base area as the cube. The height of the cube is \( s \), and the base area is \( s^2 \).
3. **Volume of the pyramid**: The volume of a pyramid is calculated using the formula:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Since the base area of the pyramid is \( s^2 \) and the height is \( s \), we can express the volume of the pyramid in terms of \( s \):
\[
V_{\text{pyramid}} = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3
\]
4. **Substituting the volume of the cube**: We already know \( s^3 = 9 \), so:
\[
V_{\text{pyramid}} = \frac{1}{3} \times 9 = 3 \text{ cubic meters}
\]
Thus, the volume of the pyramid is:
\[
\boxed{3 \, \text{m}^3}
\]
1. **Volume of the cube**: The volume of a cube is calculated using the formula:
\[
V_{\text{cube}} = s^3
\]
where \( s \) is the side length of the cube. Given that the volume of the cube is 9 cubic meters, we can set up the equation:
\[
s^3 = 9
\]
To find \( s \), take the cube root:
\[
s = \sqrt[3]{9}
\]
2. **Height and base area of the pyramid**: The pyramid has the same height and base area as the cube. The height of the cube is \( s \), and the base area is \( s^2 \).
3. **Volume of the pyramid**: The volume of a pyramid is calculated using the formula:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Since the base area of the pyramid is \( s^2 \) and the height is \( s \), we can express the volume of the pyramid in terms of \( s \):
\[
V_{\text{pyramid}} = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3
\]
4. **Substituting the volume of the cube**: We already know \( s^3 = 9 \), so:
\[
V_{\text{pyramid}} = \frac{1}{3} \times 9 = 3 \text{ cubic meters}
\]
Thus, the volume of the pyramid is:
\[
\boxed{3 \, \text{m}^3}
\]