Question

Suppose a sphere, cylinder, and cone all share the same radius and the cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone? (1 point)
• sphere - cone = cylinder
• 3 cones = 1 cylinder
• sphere = cylinder + cone
• 1 cylinder = 2 cones + 1 sphere

Answer 1

To analyze the relationship between the volumes of a sphere, cylinder, and cone, we can start by calculating the volumes based on the given dimensions:

1. **Definitions:**
- Let the radius \( r \) be the same for all shapes.
- The height \( h \) of the cylinder and cone is \( 2r \).

2. **Volume Formulas:**
- **Volume of the sphere (\( V_s \)):**
\[
V_s = \frac{4}{3} \pi r^3
\]
- **Volume of the cylinder (\( V_c \)):**
\[
V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]
- **Volume of the cone (\( V_{cone} \)):**
\[
V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

3. **Now, we can examine the relationships:**
- **1. sphere - cone = cylinder:**
\[
V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \quad \text{(not equal to } V_c)
\]
- **2. 3 cones = 1 cylinder:**
\[
3 V_{cone} = 3 \cdot \left(\frac{2}{3} \pi r^3\right) = 2 \pi r^3 = V_c \quad \text{(this is true)}
\]
- **3. sphere = cylinder + cone:**
\[
V_s = V_c + V_{cone} \Rightarrow \frac{4}{3} \pi r^3 = 2\pi r^3 + \frac{2}{3} \pi r^3 \quad \text{(not true)}
\]
- **4. 1 cylinder = 2 cones + 1 sphere:**
\[
V_c = 2 V_{cone} + V_s \Rightarrow 2 V_{cone} + V_s = 2 \cdot \frac{2}{3} \pi r^3 + \frac{4}{3} \pi r^3 \Rightarrow \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \quad \text{(not true)}
\]

The only true statement is:

**3 cones = 1 cylinder**.