Question

Suppose a sphere, cylinder, and 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

answers:

sphere = cylinder + cone

3 cones = 1 cylinder

1 cylinder = 2 cones + 1 sphere

sphere - cone = cylinder

Answer 1

To analyze the relationship between the volumes of the sphere, cylinder, and cone, we can use their respective volume formulas:

1. **Volume of the sphere (V_s)**:
\[
V_s = \frac{4}{3} \pi r^3
\]

2. **Volume of the cylinder (V_c)**:
\[
V_c = \pi r^2 h
\]

3. **Volume of the cone (V_cone)**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]

Assuming the cylinder and cone have the same height \( h \), we can express the volumes using that variable.

Now, let's evaluate each of the proposed relationships:

1. **sphere = cylinder + cone**:
This implies:
\[
\frac{4}{3} \pi r^3 = \pi r^2 h + \frac{1}{3} \pi r^2 h
\]
This is not generally true unless specific values of \( r \) and \( h \) hold.

2. **3 cones = 1 cylinder**:
This implies:
\[
3 \left(\frac{1}{3} \pi r^2 h\right) = \pi r^2 h
\]
This is true.

3. **1 cylinder = 2 cones + 1 sphere**:
This implies:
\[
\pi r^2 h = 2\left(\frac{1}{3} \pi r^2 h\right) + \frac{4}{3} \pi r^3
\]
This is not true in general.

4. **sphere - cone = cylinder**:
This implies:
\[
\frac{4}{3} \pi r^3 - \frac{1}{3} \pi r^2 h = \pi r^2 h
\]
This is also not true in general.

Based on this analysis, the **correct relationship** among those proposed is:

**3 cones = 1 cylinder**.