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

sphere = cylinder + cone
sphere = cylinder + cone

cone = cylinder − sphere
cone = cylinder − sphere

cone = sphere − cylinder
cone = sphere − cylinder

cylinder = sphere − cone

Answer 1

To find the correct relationship between the volumes of the sphere, cylinder, and cone when they share the same radius (let's denote it as \( r \)) and the cylinder and cone share the same height (denote it as \( h = 2r \)), we need to calculate the volumes of each shape.

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_{cone})**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now let's examine the relationships:

- We already have:
\[
V_c = 2\pi r^3
\]
\[
V_{cone} = \frac{2}{3} \pi r^3
\]

Next, we can combine the volumes of the cone and sphere:

Adding the volume of the cone to the volume of the sphere:
\[
V_s + V_{cone} = \frac{4}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 = 2\pi r^3 = V_c
\]

Thus, the relationship established is:
\[
V_s + V_{cone} = V_c
\]
or rearranging,
\[
V_c = V_s + V_{cone}
\]

This leads to the conclusion that:
- **Cylinder = Sphere + Cone**.

Therefore, the correct response is:

**sphere = cylinder + cone**