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?
0.375
Radius of the conical vessel, R=AC=6cm
Height of the conical vessel, h=OC=8cm
Radius of the sphere, PD=PC=r
∴PC=PD=rAC=AD=6 cm
[Since, lengths of two tangents from an external point to a circle are equal]
△OCA & △OPD are right triangle.
[∵ Tangent and radius are perpendicular to each other]
OA=√OC2+AC2=√82+62 =√100=10 cmOP2=OD2+PD2
OD=OA−AD=10−6=4 cmOP=OC−PC=8−r (8−r)2=42+r264−16r+r2=16+r216r=48⇒r=3 cm.
Volume of water overflown = Volume
of sphere
=43πr3=43π×(3)3=36π cm3
Original volume of water = volume of
cone
=13πr2h=13π×62×8=96π cm3
∴ Fraction of water overflown =Volume of water overflownOriginal volume of water=36π96π=38=0.375
∴ Fraction of water overflown is 0.375