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Relative Error of Right Triangle Area

  1. Jan 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Anyone mind checking my answer for me?

    The area of a right triangle with hypotenuse H is

    A=1/4H^2sin(2x)

    x is one of the acute angles. Use differentials to estimate the relative errors of the area if H = 4cm exactly and x is measured to be 30degrees with an error of measurement of 15 minutes of arc.


    2. Relevant equations

    A=1/4H^2sin(2x)
    dA=1/2H^2cos(2x)dx
    dA/A=2cot(60)(0.25)

    3. The attempt at a solution
    A=1/4*16sin(60)=3.464cm^2
    dA=8cos(60)(0.25)
    =0.125cm^2
    dA/A=0.125/3.464=0.2887=relative error

    Now, did I have to convert to radians in step one? 30 to pi/6? Or is this right?
     
  2. jcsd
  3. Jan 18, 2012 #2

    Dick

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    The formula d(sin(2x))=2*cos(2x)*dx is correct only if x is measured in radians. You can call the 2x part either 60 degrees or pi/3 if you adjust your calculator for degrees or radians, but the dx part definitely needs to be in radians.
     
  4. Jan 18, 2012 #3
    Thanks Dick. So the steps I took were correct just needed to convert to radians?

    How does this look?
    15/60 degrees = 0.00436rad=dx
    A=1/4*16sin(pi/3)=3.464cm^2
    dA=8cos(pi/3)(0.00436)
    =0.01744cm^2
    dA/A=0.01744/3.464≈0.005=relative error
     
  5. Jan 18, 2012 #4

    Dick

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    That sounds much better.
     
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