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

sphere = cylinder + cone
sphere = cylinder + cone

cone = cylinder − sphere
cone = cylinder − sphere

cone = sphere − cylinder

Answer 1

To analyze the relationship between the volumes of the sphere, cylinder, and cone, let's start by calculating their respective volumes using the given dimensions.

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

2. **Cylinder**: The volume of a cylinder is given by:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
Given that the height \( h \) is twice the radius \( r \):
\[
h = 2r \quad \Rightarrow \quad V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3
\]

3. **Cone**: The volume of a cone is given by:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Again, using \( h = 2r \):
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now, we can compare the volumes:

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

Now, let's check the relationships between the volumes:

1. **Check \( V_{\text{cylinder}} = V_{\text{sphere}} - V_{\text{cone}} \)**:
\[
V_{\text{sphere}} - V_{\text{cone}} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
This is equal to \( V_{\text{cone}} \), not \( V_{\text{cylinder}} \).

2. **Check \( V_{\text{sphere}} = V_{\text{cylinder}} + V_{\text{cone}} \)**:
\[
V_{\text{cylinder}} + V_{\text{cone}} = 2\pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]
This does not equal \( V_{\text{sphere}} \).

3. **Check \( V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}} \)**:
\[
V_{\text{cylinder}} - V_{\text{sphere}} = 2\pi r^3 - \frac{4}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
This is equal to \( V_{\text{cone}} \).

4. **Check \( V_{\text{cone}} = V_{\text{sphere}} - V_{\text{cylinder}} \)**:
\[
V_{\text{sphere}} - V_{\text{cylinder}} = \frac{4}{3} \pi r^3 - 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{6}{3} \pi r^3 = -\frac{2}{3} \pi r^3
\]
This does not equal \( V_{\text{cone}} \).

From this analysis:

The correct relationship is:
\[
V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}}
\]

Therefore, the correct response is:
**cone = cylinder − sphere**.