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Math Help - [SOLVED] error propagation: the approximation formula

  1. #1
    Member sinewave85's Avatar
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    [SOLVED] error propagation: the approximation formula

    I can't seem to get the right answer to this one, and I hope someone can tell me what I am doing wrong!

    "You measure the radius of a circle to be 12 cm and use the formula A = pi(r^2) to calculate the area. If your measurement of the radius is accurate to within 3%, approximately how accurate (to the nearest percent) is your calculation of the area?

    A(r) = \pi r^2
    A'(r) = 2\pi r
    \left|\frac{dA}{A}\right|= \left|\frac{2 \pi r dr}{ \pi r^2}\right|
    \left|\frac{dA}{A}\right| = 2\left|\frac{dr}{r}\right|
    \left|\frac{dA}{A}\right| = 2\left|\frac{\pm0.03(10)}{12}\right|
    \left|\frac{dA}{A}\right| = 0.05

    The problem is that the answer given in the back of the book is 0.06. So, where did I mess up? Thanks for the help!
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  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
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    Hi
    Quote Originally Posted by sinewave85 View Post
    I can't seem to get the right answer to this one, and I hope someone can tell me what I am doing wrong!

    "You measure the radius of a circle to be 12 cm and use the formula A = pi(r^2) to calculate the area. If your measurement of the radius is accurate to within 3%, approximately how accurate (to the nearest percent) is your calculation of the area?

    A(r) = \pi r^2
    A'(r) = 2\pi r
    \left|\frac{dA}{A}\right|= \left|\frac{2 \pi r dr}{ \pi r^2}\right|
    \left|\frac{dA}{A}\right| = 2\left|\frac{dr}{r}\right|
    Taking it from here, the relative error of r is \left|\frac{\mathrm{d}r}{r}\right|=0.03 so \left|\frac{\mathrm{d}A}{A}\right|=2\times 0.03=0.06. Be careful to make a difference between absolute error ( |\mathrm{d}r| ) and relative error. \left( \left|\frac{\mathrm{d}r}{r}\right| \right)
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