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Calculating the uncertainty in an A4 page

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Calculate the uncertainty in the area of an A4 page using a 30cm ruler with mm divisions.


    2. Relevant equations



    3. The attempt at a solution


    Page is measured as:

    Height H = 297mm

    Width W = 210mm

    H x W = 62370 mm^2



    I'm using this formula to calculate the area uncertainty:

    (dA/A)^2 = (dH/H)^2 + (dW/W)^2



    (0.5/297)^2 + (0.5/210)^2 = (dA/62370)^2

    (0.000002834 + 0.000005669)^0.5 = dA/62370

    (0.000008503)^0.5 = dA/62370

    0.002916 = dA/62370

    dA = 181.87



    So the area is:

    62370 +/- 180 mm^2



    Have I done this correctly?



    Thank you.
     
  2. jcsd
  3. Sep 20, 2012 #2
    You could have simplified your calculation if you observed that ## (\frac {dH} {H})^2 + (\frac {dW} {W})^2 = (\frac {dA} {A})^2 = (\frac {dA} {HW})^2 ##, so you can multiply the equation by ##(HW)^2 ##, getting ## W^2dH^2 + H^2dW^2 = dA^2 ##. This has a very simple geometric interpretation, as ##WdH ## and ## HdW ## are the areas of rectangular margins at the sides of the paper during to uncertainty. Further noting that ## dH = dW = dx = 0.5 ##, you get ## dA = dx \sqrt {H^2 + W^2} ##.
     
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