MathematicsA rectangular sheet of paper 30 cm×18 cm   be transformed into the curved surface of a right circular cylinder in two ways i.e., either by rolling the paper along its length or by rolling it along its breadth. What is the ratio of the volumes of the two cylinders thus formed?

A rectangular sheet of paper 30 cm×18 cm   be transformed into the curved surface of a right circular cylinder in two ways i.e., either by rolling the paper along its length or by rolling it along its breadth. What is the ratio of the volumes of the two cylinders thus formed?


  1. A
    2:5
  2. B
    5:2
  3. C
    3:5
  4. D
    5:3 

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    Solution:

    It is given that,
    The size of the paper is 30 cm by 18 cm.
    (Cylinder 1) By rolling lengthwise.
    Height of cylinder 1 (h1) = 18 cm  , diameter of the cylinder 1 = 30 cm
    Since the circumference of the base of the cylinder is 2πr  . So,  2π r 1 =30 r 1 = 30 2π r 1 = 30×7 2×22 r 1 = 105 22  cm    Thus, the volume (V1) is,
    =π r 1 2 h 1 = 22 7 × 105 22 × 105 22 ×18c m 3    (Cylinder II) By rolling breadthwise.  Height of cylinder 2 ( h 2 )=30cm  , diameter of cylinder 2 = 18 cm  
    Since the circumference of the base of the cylinder is 2πr  . So,
    2π r 2 =18 r 2 = 18 2π r 2 = 63 22  cm    Thus, the volume (V2) is,
    =π r 2 2 h 2 = 22 7 × 63 22 × 63 22 ×30c m 3    Now, divide V 1   by V 2  .
    V1V2=227×10522×10522×18227×6322×6322×30
    =105×105×1863×63×30
    =198450119070
    =53      Thus, the ratio of the two cylinders is 5:3  .
    Therefore, option 4 is correct.
     
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