Question

Which best describes the relationship among the volumes of hemisphere X, cylinder Y, and cone Z?

Answer 1

To describe the relationship among the volumes of a hemisphere (X), a cylinder (Y), and a cone (Z), we can start by recalling their volume formulas:

1. **Volume of a hemisphere (X)**: The formula for the volume of a hemisphere with radius \( r \) is:
\[
V_X = \frac{2}{3} \pi r^3
\]

2. **Volume of a cylinder (Y)**: The volume of a cylinder with radius \( r \) and height \( h \) is:
\[
V_Y = \pi r^2 h
\]

3. **Volume of a cone (Z)**: The volume of a cone with radius \( r \) and height \( h \) is:
\[
V_Z = \frac{1}{3} \pi r^2 h
\]

Now, we can analyze the relationships between these volumes under certain conditions.

Assuming that the hemisphere, cylinder, and cone all share the same radius \( r \) and height \( h \):

- The volume of the cone \( V_Z \) is one-third of the volume of the cylinder \( V_Y \):
\[
V_Z = \frac{1}{3} V_Y
\]

- The hemisphere shares the same radius but has a different volume. If the height of the cylinder (and cone) is equal to the diameter of the hemisphere (i.e., \( h = 2r \)), we can express volumes with respect to a shared parameter.

From the equation, we can generally conclude:

1. The cone's volume is exactly one-third of the cylinder's volume.
2. The relationship with the hemisphere will depend on the height choice. If you set up conditions suitably, you may find that the volumes can be compared accordingly.

In summary:
- If \( h = 2r \), then
\[ V_Y = \pi r^2 (2r) = 2 \pi r^3 \]
So, \( V_Z = \frac{1}{3} V_Y \) and \( V_X = \frac{2}{3} \pi r^3 \).

To summarize:
- **Volume relationships** will depend on the height relative to the radius.
- Generally, \( V_Z = \frac{1}{3} V_Y \) and need to relate \( V_X \) based on conditions.