MathematicsThe height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volumes of the two parts.

The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volumes of the two parts.


  1. A
    1:7
  2. B
    7:1
  3. C
    2:7
  4. D
    7:2 

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    Solution:

    Given,
    Height of cone is 10cm.
    According to the question, the figure will be,
    Here, the upper part is a cone and the lower part is a frustum.
    Now, in the figure
    In triangle OCD and OAB,
    COD=AOB    [Same angle]
    OCD=OAB    [Right angles]
    ODC=OBA    [Corresponding angles]
    Thus, CODOAB  
    So,
    OC OA = CD AB h 10 = r R    Now, the height of the smaller cone is half the height of the original cone.
    So, h=5  
    We get,
    5 10 = r R 1 2 = r R R=2r   The volume of a cone of radius r and height h is given by V c = 1 3 π r 2 h  .
    V c = 1 3 π r 2 5 V c = 5 3 π r 2  
    Thus, the volume of upper part is 5 3 π r 2 c m 3  
    The volume of the frustum of cone is given by πh 3 R 2 + r 2 +Rr  
    We get,
    V F = π 5 3 2r 2 + r 2 + 2r r V F = 5π 3 4 r 2 + r 2 +2 r 2 V F = 5π 3 7 r 2 V F = 35 3 π r 2  
    Thus, the volume of lower part is 35 3 π r 2 c m 3  .
    The ratio of the volume of the upper part to the volume of the lower part of the original cone is given by,
    V c V F = 5 3 π r 2 35 3 π r 2 V c V F = 5 35 V c V F = 1 7   Thus, the ratio of the volumes of two parts is 1:7.
    Hence, the correct option is 1
     
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