Question

Suppose a sphere, cylinder, and cone all share the same radius. 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)
Responses

cylinder = sphere – cone
cylinder = sphere – cone

cone = sphere – cylinder
cone = sphere – cylinder

sphere = cylinder + cone
sphere = cylinder + cone

cone = cylinder – sphere

Answer 1

To determine the correct relationship between the volumes of the sphere, cylinder, and cone, we first need to calculate their volumes using their respective formulas.

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

2. **Volume of the Cylinder (V_c)**:
The height (h) of the cylinder is twice the radius (r), so \( h = 2r \).
\[
V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]

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

Now we can compare the volumes:

- Volume of the Sphere:
\[
V_s = \frac{4}{3} \pi r^3
\]

- Volume of the Cylinder:
\[
V_c = 2\pi r^3
\]

- Volume of the Cone:
\[
V_{cone} = \frac{2}{3} \pi r^3
\]

Next, we can rewrite \(V_{c}\) and \(V_{cone}\) to express them in terms of \(V_s\):

1. **Expressing the Cylinder in terms of the Sphere and Cone**:
\[
V_c = V_s - V_{cone}
\]

2. **Substituting volumes**:
- \(V_s = \frac{4}{3} \pi r^3\)
- \(V_{cone} = \frac{2}{3} \pi r^3\)

So:
\[
V_c = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
This means the formula \(V_c = V_s - V_{cone}\) holds true.

Now let's examine each response option:

1. **cylinder = sphere – cone**: (This is true)
2. **cone = sphere – cylinder**: (This is false)
3. **sphere = cylinder + cone**: (This is false)
4. **cone = cylinder – sphere**: (This is false)

Thus, the correct relationship is:

**cylinder = sphere – cone**