A rectangular sheet of paper 30cm * 18cm can be transformed into the curved surface of a right circular cylinder in two ways by rolling the paper along its length or by rolling it along its breadth.find the ratio of the volumes of the two cylinders thus formed.
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GIVEN :
DIMENSIONS OF A RECTANGULAR SHEET :
LENGTH :
=> 30 CM
BREADTH :
=> 18 CM
_____________________________
CONDITION ;
=> IT IS ROLLED ALONG ITS LENGTH :
______________________________
=> FROM THE ATTACHMENT YOU WILL SEE ,
=> HOW THE CYLINDER IS FORMED ...
_____________________________
NOW ,
IT IS ROLLED ALONG THE LENGTH,
=> THE CIRCUMFERENCE OF THE BASE OF THE CYLINDER WHICH IS CIRCULAR
=> IS THE LENGTH OF THAT SHEET :
______________________________
SO,
=> CIRCUMFERENCE OF THE BASE :
=> 30 CM
______________________________
STEP 1 :
=> FROM THE FORMULA OF CIRCUMFERENCE OF A CIRCLE FIND THE RADIUS OF THAT CYLINDER :
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FORMULA :
=> CIRCUMFERENCE OF A CIRCLE :
=> 2 * π * R
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FROM ALL THE INFO ,
=> 2 * π * R = LENGTH OF SHEET
=> 2 * 22 / 7 * R = 30 CM
=> 2 * 22 * R = 30 * 7
=> R = 30 * 7 / 2 * 22
=> R = 15 * 7 / 22
=> R = 105 / 22
=> R = 4.7727... CM
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STEP 2 :
=> TO FIND VOLUME OF THE CYLINDER :
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VOLUME OF A CYLINDER :
=> π * R^2 * H
_____________________________
HEIGHT OF THE CYLINDER :
=> IS THE BREADTH OF THE SHEET
=> SEE FROM FIG.
_____________________________
VOLUME :
=> 22 / 7 * 4.7 * 4.7 * 18 CM^3
=> 15 * 4.7 * 18
=> 1287.9 CM^3 .....
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CONDITION 2 :
=> IF THE SHEET IS ROLLED ALONG ITS BREADH
______________________________
STEP 3 :
=> FIND THE RADIUS USING THE FORMULA FOR CIRCUMFERENCE OF A CIRCLE
_____________________________
2 * π * R = 18 CM
=> π * R = 18 / 2
=> π * R = 9
=> 22 / 7 * R = 9
=> R = 7 * 9 / 22
=> R = 63 / 22
=> R = 2.8636 CM
____________________________
STEP 4 :
=> FIND THE VOLUME OF THE CYLINDER
_____________________________
FORMULA :
=> π * R^2 * H
_____________________________
=> HEIGHT OF THIS CYLINDER IS THE LENGTH OF THE SHEET
______________________________
π * R^2 * H
=> 9 * R * 30
=> 270 * 2.86
=> 772.2 CM^3
_____________________________
STEP 5 :
=> FIND THE RATIO OF THE TWO VOLUMES :
______________________________
RATIO :
=> 1287.9 / 772.2
=> 12879 / 7722
=> 4293 / 2574
=> 1431 / 858
=> 477 / 286
_______________________________
SO ,
=> RATIO :
=> 477 : 286
_______________________________
GIVEN :
DIMENSIONS OF A RECTANGULAR SHEET :
LENGTH :
=> 30 CM
BREADTH :
=> 18 CM
_____________________________
CONDITION ;
=> IT IS ROLLED ALONG ITS LENGTH :
______________________________
=> FROM THE ATTACHMENT YOU WILL SEE ,
=> HOW THE CYLINDER IS FORMED ...
_____________________________
NOW ,
IT IS ROLLED ALONG THE LENGTH,
=> THE CIRCUMFERENCE OF THE BASE OF THE CYLINDER WHICH IS CIRCULAR
=> IS THE LENGTH OF THAT SHEET :
______________________________
SO,
=> CIRCUMFERENCE OF THE BASE :
=> 30 CM
______________________________
STEP 1 :
=> FROM THE FORMULA OF CIRCUMFERENCE OF A CIRCLE FIND THE RADIUS OF THAT CYLINDER :
______________________________
FORMULA :
=> CIRCUMFERENCE OF A CIRCLE :
=> 2 * π * R
_____________________________
FROM ALL THE INFO ,
=> 2 * π * R = LENGTH OF SHEET
=> 2 * 22 / 7 * R = 30 CM
=> 2 * 22 * R = 30 * 7
=> R = 30 * 7 / 2 * 22
=> R = 15 * 7 / 22
=> R = 105 / 22
=> R = 4.7727... CM
______________________________
STEP 2 :
=> TO FIND VOLUME OF THE CYLINDER :
______________________________
VOLUME OF A CYLINDER :
=> π * R^2 * H
_____________________________
HEIGHT OF THE CYLINDER :
=> IS THE BREADTH OF THE SHEET
=> SEE FROM FIG.
_____________________________
VOLUME :
=> 22 / 7 * 4.7 * 4.7 * 18 CM^3
=> 15 * 4.7 * 18
=> 1287.9 CM^3 .....
_____________________________
CONDITION 2 :
=> IF THE SHEET IS ROLLED ALONG ITS BREADH
______________________________
STEP 3 :
=> FIND THE RADIUS USING THE FORMULA FOR CIRCUMFERENCE OF A CIRCLE
_____________________________
2 * π * R = 18 CM
=> π * R = 18 / 2
=> π * R = 9
=> 22 / 7 * R = 9
=> R = 7 * 9 / 22
=> R = 63 / 22
=> R = 2.8636 CM
____________________________
STEP 4 :
=> FIND THE VOLUME OF THE CYLINDER
_____________________________
FORMULA :
=> π * R^2 * H
_____________________________
=> HEIGHT OF THIS CYLINDER IS THE LENGTH OF THE SHEET
______________________________
π * R^2 * H
=> 9 * R * 30
=> 270 * 2.86
=> 772.2 CM^3
_____________________________
STEP 5 :
=> FIND THE RATIO OF THE TWO VOLUMES :
______________________________
RATIO :
=> 1287.9 / 772.2
=> 12879 / 7722
=> 4293 / 2574
=> 1431 / 858
=> 477 / 286
_______________________________
SO ,
=> RATIO :
=> 477 : 286
_______________________________
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