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 3 cones = 1 cylinder 3 cones = 1 cylinder sphere = cylinder + cone sphere = cylinder + cone sphere - cone = cylinder sphere - cone = cylinder 1 cylinder = 2 cones + 1 sphere

Answer 1

Let's analyze the volumes of the sphere, cylinder, and cone based on the given conditions.

Let the radius be \( r \) and the height of the cylinder and cone be \( h = 2r \).

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

2. **Volume of the Cylinder (V_c)**:
\[
V_c = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3
\]

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

Now, let's compare these volumes to see which equations hold true:

- **Three Cones Compared to One Cylinder**:
\[
3 V_co = 3 \left(\frac{2}{3} \pi r^3\right) = 2 \pi r^3 = V_c
\]
\( 3 \) cones do equal \( 1 \) cylinder.

- **Sphere vs. Cylinder and Cone**:
Combine the volumes of the cylinder and cone:
\[
V_c + V_co = (2 \pi r^3) + \left(\frac{2}{3} \pi r^3\right) = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]
This does not equal the sphere's volume.

- **Sphere Minus Cone Compared to Cylinder**:
\[
V_s - V_co = \frac{4}{3} \pi r^3 - \left(\frac{2}{3} \pi r^3\right) = \frac{2}{3} \pi r^3
\]
This does not equal the volume of the cylinder since \( V_c = 2 \pi r^3 \).

- **Equation Relating Cylinder, Cones, and Sphere**:
We need to know whether:
\[
1 V_c = 2 V_co + V_s
\]
Since \( V_c = 2 \pi r^3, \) \( V_co = \frac{2}{3} \pi r^3, \) and \( V_s = \frac{4}{3} \pi r^3, \)
we can rewrite this as:
\[
2V_co + V_s = 2\left(\frac{2}{3} \pi r^3\right) + \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]
This does not equal \( V_c \).

### Conclusion

The relationship that holds true is:
**3 cones = 1 cylinder.**

This is the correct response.