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
1 m3
1 m cubed
30 m3
Responses
6 m3
6 m cubed
3 m3
3 m cubed
1 m3
1 m cubed
30 m3
Answers
GPT-4o mini
To find the volume of the square-based pyramid, we first need to understand the relationship between the cube and the pyramid in terms of their dimensions and volumes.
1. **Volume of the Cube**: The volume of the cube is given as 9 cubic meters. The formula for the volume of a cube is:
\[
V = s^3
\]
where \( s \) is the length of one side of the cube. We can find \( s \) by taking the cube root of the volume:
\[
s = \sqrt[3]{9} \approx 2.08 \text{ meters}
\]
2. **Height and Base Area of the Pyramid**: From the problem, we know that the pyramid has the same height and base area as the cube.
- The height \( h \) of the pyramid is equal to the side length \( s \) of the cube, hence:
\[
h = 2.08 \text{ meters}
\]
- The base area \( A \) of the pyramid is equal to the area of the base of the cube, which is \( s^2 \):
\[
A = s^2 = 9 \text{ square meters}
\]
3. **Volume of the Square-Based Pyramid**: The volume \( V \) of a square-based pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the values we have:
\[
V = \frac{1}{3} \times 9 \text{ m}^2 \times 2.08 \text{ m} \approx \frac{1}{3} \times 18.72 \text{ m}^3
\]
To compute this:
\[
V \approx 6.24 \text{ m}^3
\]
Given that we're looking for the closest approximation among the answer choices and since the volumes are often rounded to significant figures, the volume of the pyramid is approximately equal to **6 cubic meters**.
Thus, the correct answer is:
- 6 m³ (or 6 m cubed).
1. **Volume of the Cube**: The volume of the cube is given as 9 cubic meters. The formula for the volume of a cube is:
\[
V = s^3
\]
where \( s \) is the length of one side of the cube. We can find \( s \) by taking the cube root of the volume:
\[
s = \sqrt[3]{9} \approx 2.08 \text{ meters}
\]
2. **Height and Base Area of the Pyramid**: From the problem, we know that the pyramid has the same height and base area as the cube.
- The height \( h \) of the pyramid is equal to the side length \( s \) of the cube, hence:
\[
h = 2.08 \text{ meters}
\]
- The base area \( A \) of the pyramid is equal to the area of the base of the cube, which is \( s^2 \):
\[
A = s^2 = 9 \text{ square meters}
\]
3. **Volume of the Square-Based Pyramid**: The volume \( V \) of a square-based pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the values we have:
\[
V = \frac{1}{3} \times 9 \text{ m}^2 \times 2.08 \text{ m} \approx \frac{1}{3} \times 18.72 \text{ m}^3
\]
To compute this:
\[
V \approx 6.24 \text{ m}^3
\]
Given that we're looking for the closest approximation among the answer choices and since the volumes are often rounded to significant figures, the volume of the pyramid is approximately equal to **6 cubic meters**.
Thus, the correct answer is:
- 6 m³ (or 6 m cubed).