A sphere of solid material of specific gravity `8` has a concentric spherical cavity and just sinks in water. The ratio of radius of cavity to that of outer radius of the sphere must be

Options

A. `(7^(1//3))/(2)`
B. `(5^(1//3))/(2)`
C. `(9^(1//3))/(2)`
D. `(3^(1//3))/(2)`

Answer

A

Explanation

Let `rho` the density of the material `rho_(0)` be the density of water. When the sphere has just started sinking the weight of the sphere`=` weight of water displaced (approx)
`implies(4)/(3)pi(R^(3)-r^(3))rhog=(4)/(3)piR^(3)rho_(0)g`
`implies(R^(3)-r^(3))rho=R^(3)rho_(0)implies((R^(3)-r^(3)))/(R^(3))=(rho_(0))/(rho)`
`implies(r)/(R)=((7)^(1//3))/(2)`