Differentials and tolerances

[OCTOPODES]

Homework Statement

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?

Homework Equations

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

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.

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

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

We replace \Delta V in this inequality by its approximation

dV=\left(\dfrac{dV}{dl}\right)dl=5\pi l \ dl.

This gives

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

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

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I need some clarification for this solution. Could somebody annotate it, or perhaps write up a more intuitive one?

 

[haruspex]

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.