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 dnesity 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)) rho g =3/4pi R ^(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)`