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)
Responses

sphere = cylinder + cone
sphere = cylinder + cone

3 cones = 1 cylinder
3 cones = 1 cylinder

1 cylinder = 2 cones + 1 sphere
1 cylinder = 2 cones + 1 sphere

sphere - cone = cylinder
sphere - cone = cylinder

Answer 1

To determine the correct relationship between the volumes of the sphere, cylinder, and cone, let's first calculate the volumes of each shape using the given parameters.

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

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

3. **Comparison of Volumes**:
- Now, let's evaluate the relationships:
- **Sphere vs Cylinder and Cone**:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
\[
V_{\text{cylinder}} + V_{\text{cone}} = 2 \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]
Here, \( V_{\text{sphere}} \neq V_{\text{cylinder}} + V_{\text{cone}} \).

- **Comparing 3 cones and 1 cylinder**:
\[
3 \times V_{\text{cone}} = 3 \times \frac{2}{3} \pi r^3 = 2 \pi r^3 = V_{\text{cylinder}}
\]
Thus, \( 3 \text{ cones} = 1 \text{ cylinder} \).

- **Comparing 1 cylinder with 2 cones and 1 sphere**:
\[
1 \text{ cylinder} = 2 \text{ cones} + 1 \text{ sphere}
\]
Let's check:
\[
2 \times \frac{2}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]
Thus, the equality does not hold.

- **Comparing sphere and cone to cylinder**:
\[
V_{\text{sphere}} - V_{\text{cone}} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \text{ which does not equal } V_{\text{cylinder}} (= 2 \pi r^3).
\]

The only true relationship we have derived is:

**3 cones = 1 cylinder**

Thus, the response that is true about the relationship is:
- **3 cones = 1 cylinder**.