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 ${\mathrm{\rho }}_{\mathrm{c}}$ is the relative density of the
material of the shell with respect to water, then the correct statement is that the shell is

Total volume inside the shell is occupied by Air, Water, Material of shell

$\text{ let }{\mathrm{V}}_{\mathrm{w}}\text{ be volume occupied by water in shell }$

${\mathrm{V}}_{\mathrm{a}}\text{ be volume occupied by air in shell }$

${\mathrm{V}}_{\mathrm{m}}\text{ be volume occupied by material in shell }$

The Buoyant force from the water will balance the weight of water and shell.

Since the shell is half submerged. 

$\frac{{\mathrm{V}}_{\mathrm{m}}+{\mathrm{V}}_{\mathrm{a}}+{\mathrm{V}}_{\mathrm{w}}}{2}{\mathrm{\rho }}_{\mathrm{w}}\mathrm{g}={\mathrm{V}}_{\mathrm{m}}{\mathrm{\rho }}_{\mathrm{c}}{\mathrm{\rho }}_{\mathrm{w}}\mathrm{g}+{\mathrm{V}}_{\mathrm{w}}{\mathrm{\rho }}_{\mathrm{w}}\mathrm{g}$

$\Rightarrow {\mathrm{V}}_{\mathrm{w}}={\mathrm{V}}_{\mathrm{m}}\left(1-2{\mathrm{\rho }}_{\mathrm{c}}\right)+{\mathrm{V}}_{\mathrm{a}}$

${\mathrm{V}}_{\mathrm{w}}<{\mathrm{V}}_{\mathrm{a}}\Rightarrow {\mathrm{\rho }}_{\mathrm{c}}>\frac{1}{2}$

${\mathrm{V}}_{\mathrm{w}}>{\mathrm{V}}_{\mathrm{a}}\Rightarrow {\mathrm{\rho }}_{\mathrm{c}}<\frac{1}{2}$