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Math Help - determine approximate percentage error using binomial expansion

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    Question determine approximate percentage error using binomial expansion

    The shear stress τ in a shaft of diameter
    D under a torque T is given by:
    τ = kT
    πD^3 .
    Determine the approximate percentage error
    in calculating τ if T is measured 3% too
    small and D 1.5% too large.

    (answer 7.5% decrease)


    this is a question from J bird engineering mathematics book. it gives the answer but i need to know how to do the method.

    any help much appreciated
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  2. #2
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    Approximate increases using binomial expansion

    Hello decoy808
    Quote Originally Posted by decoy808 View Post
    The shear stress τ in a shaft of diameter
    D under a torque T is given by:
    τ = kT
    πD^3 .
    Determine the approximate percentage error
    in calculating τ if T is measured 3% too
    small and D 1.5% too large.

    (answer 7.5% decrease)


    this is a question from J bird engineering mathematics book. it gives the answer but i need to know how to do the method.

    any help much appreciated
    If y = kx^n, and x and y increase by small amounts \delta x and \delta y, then

    y+ \delta y = k(x+\delta x)^n

    \Rightarrow \frac{y+ \delta y}{y }= \frac{k(x+ \delta x)^n}{kx^n}

    \Rightarrow 1 + \frac{\delta y}{y}= \left(1 + \frac{\delta x}{x}\right)^n

    \approx 1 + n\frac{\delta x}{x}, ignoring higher powers of \frac{\delta x}{x}, which we may do for small increases in x

    \Rightarrow the fractional increase in y \approx n \times the fractional increase in x.

    It is very easy show that this may be extended to a formula for y which involves powers of any number of variables.

    So, using the same method you may prove that if \tau = \frac{k}{\pi}T\cdot D^{-3}, then

    \frac{\delta \tau}{\tau} \approx \frac{\delta T}{T} -3 \frac{\delta D}{D}

    With the percentage errors you have been given

    \frac{\delta T}{T} = -0.03 and \frac{\delta D}{D} = 0.015

    So \frac{\delta \tau}{\tau} \approx -0.03 -0.045 = -0.075 = -7.5%

    Grandad
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