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` be 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 4/3pi(R^(3)-r^(3))rhog=4/3piR^(3)rho_(0)g`
`implies (R^(3)-r^(3))rho=R^(3)rho0implies((R^(3)-r^(3)))/R^(3)=(rho_(0))/(rho)`
`rarr r/R=(7^(1/3))/2`