A sphere is placed in an inverted hollow conical vessel of base radius 5 cm and vertical
height 12 cm. If the highest point of the sphere is at the level of the base of the cone,
find the radius of the sphere. Show that the volume of the sphere and the conical vessel
are as 40 : 81.
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Volume of the sphere:Volume of the cone=40:81
Step-by-step explanation:
Let the radius of the sphere = r cm.
OD = OE = r
Height of the cone EA = 12 cm
Slant height AC = √{ 52 + 122 } = √169 = 13 cm.
In ΔAEC and ODA we have
∠AEC = ∠ODA = 90°
⇒ ΔAEC ≃ ODA [ AA similarity ]
⇒ OD / OA = EC / AC
⇒ r / ( 12 - r) = 5 / 13
⇒ r = 10 / 3 cm
∴ Volume of the sphere / Volume of the cone = ( 4/3)πr3 / (1/3)πr2h
= (4 x 10/3 x 10/3 x 10/3) / (5x5x12)
= 40 / 81
Volume of the sphere : Volume of the cone= 40 : 81
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