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 6m3 6m cubed 30m3 30m cubed 3 m3 3 m cubed 1m3
Answers
GPT-4o mini
To find the volume of the pyramid, we first need to understand the relationship between the cube and the pyramid.
Given that the volume of the cube is 9 cubic meters, we can use the formula for the volume of a cube, which is:
\[
V_{\text{cube}} = s^3
\]
Where \( s \) is the length of a side of the cube. We have:
\[
s^3 = 9
\]
To find \( s \), we can take the cube root of both sides:
\[
s = \sqrt[3]{9}
\]
Next, we know that the pyramid has the same height and base area as the cube. The height of the cube is also \( s \), and the base area of the cube (which is a square) is:
\[
\text{Area}_{\text{base}} = s^2
\]
The height \( h \) of the pyramid is equal to \( s \), and the base area of the pyramid is also \( s^2 \). The volume of a pyramid is given by:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the known values into the volume formula for the pyramid gives us:
\[
V_{\text{pyramid}} = \frac{1}{3} \times s^2 \times s = \frac{1}{3} \times s^3
\]
We know \( s^3 = 9 \) (from the volume of the cube), so we substitute this into the pyramid's volume equation:
\[
V_{\text{pyramid}} = \frac{1}{3} \times 9 = 3
\]
Thus, the volume of the pyramid is:
\[
\boxed{3 \text{ m}^3}
\]
Given that the volume of the cube is 9 cubic meters, we can use the formula for the volume of a cube, which is:
\[
V_{\text{cube}} = s^3
\]
Where \( s \) is the length of a side of the cube. We have:
\[
s^3 = 9
\]
To find \( s \), we can take the cube root of both sides:
\[
s = \sqrt[3]{9}
\]
Next, we know that the pyramid has the same height and base area as the cube. The height of the cube is also \( s \), and the base area of the cube (which is a square) is:
\[
\text{Area}_{\text{base}} = s^2
\]
The height \( h \) of the pyramid is equal to \( s \), and the base area of the pyramid is also \( s^2 \). The volume of a pyramid is given by:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the known values into the volume formula for the pyramid gives us:
\[
V_{\text{pyramid}} = \frac{1}{3} \times s^2 \times s = \frac{1}{3} \times s^3
\]
We know \( s^3 = 9 \) (from the volume of the cube), so we substitute this into the pyramid's volume equation:
\[
V_{\text{pyramid}} = \frac{1}{3} \times 9 = 3
\]
Thus, the volume of the pyramid is:
\[
\boxed{3 \text{ m}^3}
\]