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
6 m3
6 m cubed
1 m3
1 m cubed
30 m3
Responses
3 m3
3 m cubed
6 m3
6 m cubed
1 m3
1 m cubed
30 m3
Answers
GPT-4o mini
To solve for the volume of the square-based pyramid, we can start with the information given about 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. We know the volume of the cube is 9 cubic meters:
\[
s^3 = 9
\]
From this, we can find \( s \):
\[
s = \sqrt[3]{9} \approx 2.08 \text{ m}
\]
Next, we know the pyramid has the same height \( h \) and base area \( A \) as the cube. The base area of the cube is given by \( s^2 \):
\[
A = s^2 = (2.08)^2 \approx 4.33 \text{ m}^2
\]
Now, we can find the height \( h \) of the pyramid, which is the same as the side length of the cube:
\[
h = s \approx 2.08 \text{ m}
\]
The volume \( V \) of a square-based pyramid is given by the formula:
\[
V = \frac{1}{3} \times A \times h
\]
Substituting in the values for \( A \) and \( h \):
\[
V = \frac{1}{3} \times 4.33 \times 2.08
\]
Calculating that:
\[
V \approx \frac{1}{3} \times 9.00 \approx 3 \text{ m}^3
\]
Thus, the volume of the pyramid is:
\[
\boxed{3 \text{ m}^3}
\]
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. We know the volume of the cube is 9 cubic meters:
\[
s^3 = 9
\]
From this, we can find \( s \):
\[
s = \sqrt[3]{9} \approx 2.08 \text{ m}
\]
Next, we know the pyramid has the same height \( h \) and base area \( A \) as the cube. The base area of the cube is given by \( s^2 \):
\[
A = s^2 = (2.08)^2 \approx 4.33 \text{ m}^2
\]
Now, we can find the height \( h \) of the pyramid, which is the same as the side length of the cube:
\[
h = s \approx 2.08 \text{ m}
\]
The volume \( V \) of a square-based pyramid is given by the formula:
\[
V = \frac{1}{3} \times A \times h
\]
Substituting in the values for \( A \) and \( h \):
\[
V = \frac{1}{3} \times 4.33 \times 2.08
\]
Calculating that:
\[
V \approx \frac{1}{3} \times 9.00 \approx 3 \text{ m}^3
\]
Thus, the volume of the pyramid is:
\[
\boxed{3 \text{ m}^3}
\]