A cylindrical container of radius 6 cm and height 15 cm is filled with ice-cream. The whole ice-cream has to be distributed to 10 children in equal cones with hemispherical tops. If the height of the conical portion is 4 times the radius of its base, find the radius of the ice-cream cone.
Radius of cylindrical container = r = 6 cm
Height of cylindrical container = h = 15 cm
Volume of cylindrical container = πr2h
= 22/7 × 6 × 6 × 15 cm3
= 1697.14 cm3
Whole ice-cream has to be distributed to 10 children in equal cones with hemispherical tops.
Let the radius of hemisphere and base of cone be r’
Height of cone = h = 4 times the radius of its base
h’ = 4r’
Volume of Hemisphere = 2/3 π(r’)3
Volume of cone = 1/3 π(r’)2h’ = 1/3 π(r’)2 × 4r’
= 2/3 π(r’)3
Volume of ice-cream = Volume of Hemisphere + Volume of cone
= 2/3 π(r’)3 + 4/3 π(r’)3 = 6/3 π(r’)3
Number of ice-creams = 10
∴ total volume of ice-cream = 10 × Volume of ice-cream
= 10 × 6/3 π(r’)3 = 60/3 π(r’)3
Also, total volume of ice-cream = Volume of cylindrical container
⇒ 60/3 π(r’)3 = 1697.14 cm3
⇒ 60/3 × 22/7 × (r’)3 = 1697.14 cm3
⇒ (r’)3 = 1697.14 × 3/60 × 7/22 = 27 cm3
⇒ r = 3 cm
∴ Radius of ice-cream cone = 3 cm