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calculus

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The circumference of a sphere was measured to be 78.000 cm with a possible error of 0.50000 cm. Use linear approximation to estimate the maximum error in the calculated surface area.

Estimate the relative error in the calculated surface area.

  • calculus -

    The relative error in area is, to first approximation, twice the relative error in a linear dimension.

    That is because
    (delta A)/A = (1 + epsilon)^2 - 1
    = 1 + 2 epsilon + epsilon^2 - 1
    = 2 epsilon
    when epsilon << 1

    In your case, the answer is
    2*0.5/78 = 1/78 = 1.3%

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