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Differentials and tolerances

  1. Dec 16, 2012 #1
    1. The problem statement, all variables and given/known data

    About how accurately must the interior diameter of a 10-m high cylindrical storage tank be measured to calculate the tank's volume to within 1% of its true value?

    2. Relevant equations

    [itex]V=\frac{5}{2}\pi l^{2}[/itex], where [itex]V[/itex] is volume and [itex]l[/itex] is diameter.
    [itex]dV=5\pi l \ dl[/itex]

    3. The attempt at a solution

    I'm really uncertain as to how to go about this problem. What follows is the textbook's method for a similar problem translated into this problem's terms.

    --

    We want any inaccuracy in our measurement to be small enough to make the corresponding increment [itex]\Delta V[/itex] in the volume satisfy the inequality

    [itex]|\Delta V|\leq\frac{1}{100}V=\dfrac{\pi l^{2}}{40}[/itex].

    We replace [itex]\Delta V[/itex] in this inequality by its approximation

    [itex]dV=\left(\dfrac{dV}{dl}\right)dl=5\pi l \ dl[/itex].

    This gives

    [itex]|5\pi l\ dl|\leq\dfrac{\pi l^{2}}{40}[/itex], or [itex]|dl|\leq\dfrac{1}{5\pi l}\cdot\dfrac{\pi l^{2}}{40}=\dfrac{1}{5}\cdot\dfrac{l}{40}=0.005l[/itex].

    We should measure [itex]l[/itex] with an error [itex]dl[/itex] that is no more than 0.5% of its true value.

    --

    I need some clarification for this solution. Could somebody annotate it, or perhaps write up a more intuitive one?
     
  2. jcsd
  3. Dec 16, 2012 #2

    haruspex

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    Your answer is correct. I would tend to start from the other end. V = πl2h/4. If there's a fractional error δ in l, the computed volume will be π(l(1+δ))2h/4 = V(1+δ)2 = V + 2Vδ + Vδ2 ≈ V + 2Vδ. So the fractional error in the volume will be about double that in the linear measurement.
     
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