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

sphere = cylinder + cone

cone = cylinder – sphere

cone = sphere – cylinder

Answer 1

To analyze the relationship between the volumes of the sphere, cylinder, and cone, let's first establish their volume formulas and the given dimensions.

1. **Sphere**: The volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]

2. **Cylinder**: The volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
Since the height \( h \) of the cylinder is twice the radius \( r \), we have:
\[
h = 2r \implies V = \pi r^2 (2r) = 2\pi r^3
\]

3. **Cone**: The volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
With the same height \( h = 2r \):
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now, let's summarize the volumes:

- Volume of sphere: \( V_s = \frac{4}{3} \pi r^3 \)
- Volume of cylinder: \( V_c = 2 \pi r^3 \)
- Volume of cone: \( V_{cone} = \frac{2}{3} \pi r^3 \)

Next, we will compare these volumes to see which statement is true:

1. **Check \( V_c = V_s - V_{cone} \)**:
\[
V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 = V_{cone}
\]
This statement does not hold true.

2. **Check \( V_s = V_c + V_{cone} \)**:
\[
V_c + V_{cone} = 2 \pi r^3 + \frac{2}{3} \pi r^3 = \left(2 + \frac{2}{3}\right) \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]
This statement does not hold true either.

3. **Check \( V_{cone} = V_c - V_s \)**:
\[
V_c - V_s = 2 \pi r^3 - \frac{4}{3} \pi r^3 = \left(2 - \frac{4}{3}\right) \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 = V_{cone}
\]
This statement does hold true.

4. **Check \( V_{cone} = V_s - V_c \)**:
\[
V_s - V_c = \frac{4}{3} \pi r^3 - 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{6}{3} \pi r^3 = -\frac{2}{3} \pi r^3
\]
This statement does not hold true.

The only statement that holds true is:

**cone = cylinder – sphere.**