Question

Suppose a sphere, cylinder and cone all share the same radoius and the cylinder and cone also share the same height which of the following is true about the relationship between the volume formulas for the sheer cylinder and cone

Answer 1

To compare the volumes of a sphere, cylinder, and cone that all share the same radius (\( r \)), with the cylinder and cone also having the same height (\( h \)), we can use the formulas for each shape's volume.

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

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

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

### Relationship between the volumes of the cylinder and cone:
From the formulas for the cylinder and cone, we can see that the volume of the cone is one-third the volume of the cylinder:

\[
V_{cone} = \frac{1}{3} V_{cylinder}
\]

### Conclusion:
- The volume of the cone is one-third of the volume of the cylinder when both share the same radius and height.
- This relationship holds regardless of the specific values of \( r \) and \( h \) as long as they are consistent.

This means that if you know the volume of the cylinder, you can easily find the volume of the cone by dividing the cylinder's volume by 3.