SOLUTION: Which has greater volume . A piece of paper rolled into a cylinder Lengthwise Or .A piece of paper rolled into a cylinder widthwise

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Question 1151128: Which has greater volume
. A piece of paper rolled into a cylinder Lengthwise Or
.A piece of paper rolled into a cylinder widthwise
Found 3 solutions by josgarithmetic, MathLover1, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
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If length of the paper is y, and width of the paper is x,
x%3Cy,

and if the paper is rolled so that the circumference is y, then volume is xy%5E2%2F%284pi%29

and if paper is rolled so that circumference is x, then volume is x%5E2y%2F%284pi%29.

Answer by MathLover1(20849) About Me  (Show Source):
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Rolling the cylinder lengthwise means that the circumference of the cylinder becomes the magnitude of the width=W, and the height is the magnitude of the length=L

so, if the circumference of the cylinder is 2pi%2Ar+=+W
r+=+W%2F%282pi%29
Then the volume is:
V%5B1%5D+=+pi%2Ar%5E2%2AL
V%5B1%5D+=+pi%28W%2F%282pi%29%29%5E2%2AL
V+%5B1%5D=+pi%28W%5E2%2F%284%2Api%5E2%29%29L
V%5B1%5D+=+%281%2F%284pi%29%29W%5E2%2AL


if the circumference of the cylinder is2pi%2Ar+=+L

r+=+L%2F%282pi%29
Then the volume is:
V%5B2%5D+=+pi%2Ar%5E2%2AW
V%5B2%5D+=+pi%28L%2F%282pi%29%29%5E2%2AW
V+%5B2%5D=+pi%28L%5E2%2F%284%2Api%5E2%29%29W
V%5B2%5D+=+%281%2F%284pi%29%29L%5E2%2AW


Dividing the width folding by the length folding:


Thus the ratio of the two volumes is the ratio of the length to the width (interesting.) So the width folding has greater volume.

Answer by ikleyn(52781) About Me  (Show Source):
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.

            Start from learning on  "what a paper roll is".

            Type these words into the  GOOGLE  search engine and look into the  images  it will produce for you in response.


Solution 

Total mass is the same;


the density of the paper is the same in both cases


    ========> h e n c e =========>


the volume  MUST  be  THE  SAME  in both cases.

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Had they be different,  then the  "PERPETUUM  MOBILE"  would be possible to construct  (!)  (!)  (!)