Question
Suppose a sphere, a cylinder and a cone all share the same radius, and the cylinder and cone also share the same height.
Which of the following is true about the relationship between the volume formulas for the sphere, cylinder and cone?
-sphere=cylinder+cone
-3 cones= 1 cylinder
-1 cylinder= 2 cones+ 1 sphere
-sphere-cone= cylinder
Which of the following is true about the relationship between the volume formulas for the sphere, cylinder and cone?
-sphere=cylinder+cone
-3 cones= 1 cylinder
-1 cylinder= 2 cones+ 1 sphere
-sphere-cone= cylinder
Answers
GPT-4o mini
To analyze the relationships among the volumes of the sphere, cylinder, and cone, we first need to recall the volume formulas for each shape.
1. **Volume of a Sphere (V_s)**:
\[
V_s = \frac{4}{3} \pi r^3
\]
2. **Volume of a Cylinder (V_c)**:
\[
V_c = \pi r^2 h
\]
3. **Volume of a Cone (V_{cone})**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given that the cylinder and cone share the same height \( h \) and a radius \( r \), we can further analyze the relationships.
### Analyzing Each Statement:
1. **sphere = cylinder + cone**:
- Using the formulas:
- \( V_c + V_{cone} = \pi r^2 h + \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^2 h \).
- Since the volume of the sphere is \( \frac{4}{3} \pi r^3 \), this is not true as the left-hand side does not equal the right-hand side.
2. **3 cones = 1 cylinder**:
- Using the volumes:
- \( 3 \times V_{cone} = 3 \times \frac{1}{3} \pi r^2 h = \pi r^2 h \).
- This is true, as it equals the volume of the cylinder \( V_c = \pi r^2 h \).
3. **1 cylinder = 2 cones + 1 sphere**:
- Using the formulas:
- \( V_c = \pi r^2 h \) and \( 2 V_{cone} + V_s = 2 \times \frac{1}{3} \pi r^2 h + \frac{4}{3} \pi r^3 \).
- \( 2 V_{cone} = \frac{2}{3} \pi r^2 h \), so:
- \( 2 V_{cone} + V_s = \frac{2}{3} \pi r^2 h + \frac{4}{3} \pi r^3 \).
- This does not equal \( V_c \) in general (only if \( h = 2r \)).
4. **sphere - cone = cylinder**:
- Using the formulas:
- \( V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{1}{3} \pi r^2 h \).
- This is not generally true; \( V_c = \pi r^2 h \).
### Conclusion:
The only true statement is:
- **3 cones = 1 cylinder**.
So, the correct answer is the second statement: **3 cones = 1 cylinder**.
1. **Volume of a Sphere (V_s)**:
\[
V_s = \frac{4}{3} \pi r^3
\]
2. **Volume of a Cylinder (V_c)**:
\[
V_c = \pi r^2 h
\]
3. **Volume of a Cone (V_{cone})**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given that the cylinder and cone share the same height \( h \) and a radius \( r \), we can further analyze the relationships.
### Analyzing Each Statement:
1. **sphere = cylinder + cone**:
- Using the formulas:
- \( V_c + V_{cone} = \pi r^2 h + \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^2 h \).
- Since the volume of the sphere is \( \frac{4}{3} \pi r^3 \), this is not true as the left-hand side does not equal the right-hand side.
2. **3 cones = 1 cylinder**:
- Using the volumes:
- \( 3 \times V_{cone} = 3 \times \frac{1}{3} \pi r^2 h = \pi r^2 h \).
- This is true, as it equals the volume of the cylinder \( V_c = \pi r^2 h \).
3. **1 cylinder = 2 cones + 1 sphere**:
- Using the formulas:
- \( V_c = \pi r^2 h \) and \( 2 V_{cone} + V_s = 2 \times \frac{1}{3} \pi r^2 h + \frac{4}{3} \pi r^3 \).
- \( 2 V_{cone} = \frac{2}{3} \pi r^2 h \), so:
- \( 2 V_{cone} + V_s = \frac{2}{3} \pi r^2 h + \frac{4}{3} \pi r^3 \).
- This does not equal \( V_c \) in general (only if \( h = 2r \)).
4. **sphere - cone = cylinder**:
- Using the formulas:
- \( V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{1}{3} \pi r^2 h \).
- This is not generally true; \( V_c = \pi r^2 h \).
### Conclusion:
The only true statement is:
- **3 cones = 1 cylinder**.
So, the correct answer is the second statement: **3 cones = 1 cylinder**.