A sphere of solid material of R.D. 8 has a concentric cavity and just sinks in water. Then ratio of the radius of the cavity to that of outer radius of the sphere must be:

Options

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

Answer

C

Explanation

Let R the outer radius and r be the radius of cavity. Then by law of floatation

`4/3pir^(3)xx1xxg=4/3pi(R^(3)-r^(3))pg`
`R^(3)=(R^(3)-r^(3))p=(R^(3)-r^(3))8`
`1/8=1-(r/R)^(3)`
`(r/R)^(3)=1-1/8=7/8`
`r/R=(7/8)^(1//3)=((7)^(1//3))/2`
Correct choice is (c).