Question
Describe an informal argument for the volume of a pyramid based on the volume of a cube. A cube has a volume of 12 cubic meters. A square-based pyramid has the same base area and the same height as the cube. What is the volume of the pyramid? (1 point)
• 36 m3
• 9 m3
0 4m3
• 3 m3
• 36 m3
• 9 m3
0 4m3
• 3 m3
Answers
GPT-4o mini
To informally argue for the volume of a pyramid based on the volume of a cube, we can start by examining the relationships between the two shapes.
1. **Understanding the Cube:**
A cube has a volume calculated by the formula \( V = \text{side}^3 \). If we know that the volume of our cube is 12 cubic meters, then we can find the dimensions of the cube. Let's say the side length of the cube is \( s \). Therefore, we have:
\[
s^3 = 12 \text{ m}^3
\]
The cube has equal sides.
2. **Base of the Pyramid:**
A square-based pyramid shares the same base area as the top face of the cube. The base area of the pyramid is \( s^2 \), which equals the area of the square face of the cube (because it has the same side lengths). We can calculate the base area from the cube’s volume:
\[
\text{Base Area} = s^2
\]
3. **Height of the Pyramid:**
Since the pyramid has the same height as the cube, the height of the pyramid is \( s \), which is also the side length of the cube.
4. **Volume of the Pyramid:**
The volume of a pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the base area and height of our pyramid, we have:
\[
V = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3
\]
We already know from the volume of the cube that \( s^3 = 12 \text{ m}^3 \), so we can replace \( s^3 \):
\[
V = \frac{1}{3} \times 12 \text{ m}^3 = 4 \text{ m}^3
\]
5. **Conclusion:**
Therefore, the volume of the pyramid, which has the same base area and height as the cube, is \( 4 \) cubic meters.
From the options given, the correct answer is:
- **4 m³**.
1. **Understanding the Cube:**
A cube has a volume calculated by the formula \( V = \text{side}^3 \). If we know that the volume of our cube is 12 cubic meters, then we can find the dimensions of the cube. Let's say the side length of the cube is \( s \). Therefore, we have:
\[
s^3 = 12 \text{ m}^3
\]
The cube has equal sides.
2. **Base of the Pyramid:**
A square-based pyramid shares the same base area as the top face of the cube. The base area of the pyramid is \( s^2 \), which equals the area of the square face of the cube (because it has the same side lengths). We can calculate the base area from the cube’s volume:
\[
\text{Base Area} = s^2
\]
3. **Height of the Pyramid:**
Since the pyramid has the same height as the cube, the height of the pyramid is \( s \), which is also the side length of the cube.
4. **Volume of the Pyramid:**
The volume of a pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the base area and height of our pyramid, we have:
\[
V = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3
\]
We already know from the volume of the cube that \( s^3 = 12 \text{ m}^3 \), so we can replace \( s^3 \):
\[
V = \frac{1}{3} \times 12 \text{ m}^3 = 4 \text{ m}^3
\]
5. **Conclusion:**
Therefore, the volume of the pyramid, which has the same base area and height as the cube, is \( 4 \) cubic meters.
From the options given, the correct answer is:
- **4 m³**.