A concrete sphere of radius `R` has cavity of radius `r` which is packed with sawdust. The specific gravities of concrete and sawdust are respectively `2.4 and 0.3` for this sphere to float with its entire volume submerged under water. Ratio of mass of concrete to mass of swadust will be

Answer

`4.00`

Explanation

Let specific gravities of concrete and sawdust are `rho_(1)` and `rho_(2)` respectively.
According to principle of floatation, weight of whole sphere = upthrust on the sphere
`(4)/(3) pi (R^(3) -r^(3)) rho_(1) g +(4)/(3) pi r^(3) rho g =(4)/(3) pi R^(3) xx 1 xx g`
`rArr R^(3) rho_(1) -r^(3) rho_(1) + r^(3) rho_(2) =R^(3)`
`rArr R^(3) (rho_(1)-1) =r^(3) (rho_(1) -rho_(2)) rArr (R^(3))/(r^(3)) =(rho_(1) -rho_(2))/(rho_(1)-1)`
`rArr (R^(3) -r^(3))/( r^(3)) = (rho_(1) -rho_(2) -rho_(1) +1)/( rho_(1)-1) rArr ((R^(3) -r^(3)) rho_(1))/( r^(3)rho_(2)) =((1-rho_(2) )/(rho_(1)-1)) (rho_(1))/(rho_(2))`
`rArr ("Mass of concrete")/("Mass of saw dust") =((1-0.3)/(2.4-1)) xx (2.4)/(0.3) =4`