MathematicsA right cylindrical container of radius 6 cm   and height 15 cm   is full of ice-cream, which has to be distributed to 10 children in equal cones having hemispherical shape on the top. If the height of the conical portion is four times its base radius, find the radius of the ice-cream cone.

A right cylindrical container of radius 6 cm   and height 15 cm   is full of ice-cream, which has to be distributed to 10 children in equal cones having hemispherical shape on the top. If the height of the conical portion is four times its base radius, find the radius of the ice-cream cone.


  1. A
    3cm
  2. B
    4cm
  3. C
    5cm
  4. D
    6cm 

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    Solution:

    Given,
    Radius of cylindrical container, r = 6cm
    Height of cylindrical container, h = 15cm
    Height of the conical portion, H = 4 × radius of base of cone, x
    Let the radius of the ice-cream cone be x, then the shape of an ice-cream cone with hemispherical top is given below.
    According to question,
    Volume of cylindrical container = 10 × volume of ice cream cone
    Also,
    Volume of ice-cream cone = volume of hemispherical top + volume of cone
    Now,
    We know that the volume of a cylinder of radius, r and height, h is given by π r 2 h  .
    We get,
    V cylinder =π r 2 h V cylinder =π (6) 2 15  
    And
    The volume of a cone of radius, r and height, h is given by 1 3 π r 2 h  .
    The volume of a hemisphere of radius, r is given by 2 3 π r 2 h  .  Volume of one ice-cream cone = 2 3 π r 3 + 1 3 π r 2 h  
    We get,
     Volume of one ice-cream cone = 2 3 π r 3 + 1 3 π r 2 h Volume of one ice-cream cone= 2 3 π x 3 + 1 3 π x 2 4x Volume of one ice-cream cone = 2π x 3  
    Now,
    Volume of cylindrical container = 10 × volume of ice cream cone
    π (6) 2 15=10×2π x 3 x 3 = π (6) 2 15 10×2π x 3 = 6×6×15 10×2 x 3 =3×3×3 x=3cm  
    Therefore, the radius of the ice-cream cone is 3 cm.
    Hence, the correct option is 1.
     
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