A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If `rho_c` is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

Options

A. More than half-filled if `rho_(c)`, is less than `0.5 `
B. more than half-filled `rho_(c)` is more than `1.0 c`,
C. half filled if `rho_(c)` is more than `0.5`
D. less than half-filled if `rho_(c)` is less than `0.5`

Explanation

To solve the problem of a thin uniform cylindrical shell that is partially filled with water and floating vertically in a half-submerged state, we will analyze the forces acting on the shell and the relationship between the densities involved.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a cylindrical shell that is closed at both ends and is floating in water. It is partially filled with water and is in a vertical position, half-submerged.
- Let \( V_s \) be the volume of the shell, \( V_w \) the volume of water inside the shell, and \( V_a \) the volume of air inside the shell.

2. **Weight of the Shell**:
- The weight of the shell can be expressed as:
\[ W_{\text{shell}} = V_s \cdot \rho_c \cdot g \]
- Here, \( \rho_c \) is the density of the shell material, and \( g \) is the acceleration due to gravity.

3. **Buoyant Force**:
- The buoyant force acting on the shell is equal to the weight of the water displaced by the submerged part of the shell:
\[ F_b = (V_w + V_a) \cdot \rho_w \cdot g \]
- Here, \( \rho_w \) is the density of water.

4. **Equilibrium Condition**:
- For the shell to float in equilibrium, the weight of the shell must equal the buoyant force:
\[ W_{\text{shell}} = F_b \]
- Substituting the expressions for weight and buoyant force, we have:
\[ V_s \cdot \rho_c \cdot g = (V_w + V_a) \cdot \rho_w \cdot g \]

5. **Simplifying the Equation**:
- Canceling \( g \) from both sides, we get:
\[ V_s \cdot \rho_c = (V_w + V_a) \cdot \rho_w \]

6. **Volume Relationships**:
- Since the shell is half-submerged, we can express the volumes as:
\[ V_w + V_a = V_s \]
- This implies that the total volume of the shell is equal to the sum of the volumes of water and air.

7. **Analyzing Relative Density**:
- The relative density \( \rho_c \) is defined as the ratio of the density of the shell material to the density of water:
\[ \rho_c = \frac{\text{Density of shell}}{\text{Density of water}} \]
- If \( \rho_c < 0.5 \), it indicates that the shell is less dense than half the density of water, meaning it is more buoyant and thus more than half-filled with water.

8. **Conclusion**:
- Therefore, if \( \rho_c < 0.5 \), the shell is more than half-filled with water. If \( \rho_c > 0.5 \), it would be less than half-filled.
- The correct statement is that the shell is more than half-filled if \( \rho_c < 0.5 \).

### Final Answer:
The shell is more than half-filled if \( \rho_c < 0.5 \).