A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides it is just immersed. What fraction of water overflows ?

Explanation

By pythagoras theorem,
`aB^(2)=Ac^(2)+BC^(2)`=36+64=100
AB=10cm
Since AB is the tangent to the sphere
ODb `to` AB
`triangle` ACB `triangle` ODB
`(AC)/(Od)=(CB)/(DB)=(AB)/(OB)`
`(6)/(c )=(10)/(8-r)`
48-6r=10r `rArr` 16r=48 `rArr` r=3 cm
Volume of water in the tank=`(1)/(3)pi(6)^(2)xx8`
Water overflows due to the sphere
`therefore` volume of water overflowas=volume of sphere=`(4)/(3)pi(3)^(3)`
Required fraction of water overflows =`(4)/((3)pi(3^(3)))(1)/(1)(3)pi(6^(2))=(3)/(8)`