Question

Volume of cones cylinders and spheres quick check
Suppose a sphere cylinder and conne all share the same radius the cylinder and cone also share the same height which is twice the radius which is true about the relationship between the volume formulas for the sphere cylinder and cone

Answer 1

To analyze the relationship between the volumes of a sphere, cylinder, and cone given that they share the same radius \( r \) and that the height of the cylinder and cone is twice the radius (\( h = 2r \)), we can use the formulas for the volumes of each shape:

1. **Volume of the sphere**:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]

2. **Volume of the cylinder**:
\[
V_{cylinder} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]

3. **Volume of the 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 compare these volumes:

- **Sphere vs. Cylinder**:
- \( V_{sphere} = \frac{4}{3} \pi r^3 \)
- \( V_{cylinder} = 2 \pi r^3 \)
- The ratio of volumes:
\[
\frac{V_{sphere}}{V_{cylinder}} = \frac{\frac{4}{3} \pi r^3}{2 \pi r^3} = \frac{4/3}{2} = \frac{4}{6} = \frac{2}{3}
\]
- This tells us that the volume of the sphere is \(\frac{2}{3}\) the volume of the cylinder.

- **Cone vs. Cylinder**:
- \( V_{cone} = \frac{2}{3} \pi r^3 \)
- The ratio of volumes:
\[
\frac{V_{cone}}{V_{cylinder}} = \frac{\frac{2}{3} \pi r^3}{2 \pi r^3} = \frac{2/3}{2} = \frac{2}{6} = \frac{1}{3}
\]
- This tells us that the volume of the cone is \(\frac{1}{3}\) the volume of the cylinder.

In summary:
- The volume of the **sphere** is \(\frac{2}{3}\) the volume of the **cylinder**.
- The volume of the **cone** is \(\frac{1}{3}\) the volume of the **cylinder**.
- The volume of the **cone** is \(\frac{1}{2}\) the volume of the **sphere** when comparing them to the cylinder's volume.

Thus, the relationships can be summarized as:
\[
V_{sphere} : V_{cone} : V_{cylinder} = \frac{2}{3} : \frac{1}{3} : 1
\]