Finding the error in z

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1. Oct 12, 2014

Kyrios

1. The problem statement, all variables and given/known data

For the equation $$q = \frac{z(z+2) - 2DH}{z^2}$$ $q = -0.6 \pm 10\%$ , and z = 0.2.
D and H are known exactly.
I have to find the error in z that will give an answer of $q = -0.6 \pm 10\%$

2. Relevant equations

3. The attempt at a solution

I have considered rewriting the equation in terms of z, which gives
$$z = \frac{1 \pm \sqrt{1-2DH(q-1)}}{q-1}$$
but I'm not sure where to go with that, how the plus/minus affects it, and what to do with D and H (if anything).

I've tried to do (error in q * 0.5) / (error in q) which gives an answer for z of $0.2 \pm 10\%$ but that seems a bit too simple

2. Oct 12, 2014

ehild

It is not needed. You know that z=0.2. From that, you can determine DH. You know also q and its error.
The error of z can be obtained by differentiating both sides of the equation
$$q = \frac{z(z+2) - 2DH}{z^2}$$
$$\Delta q =\frac {dq}{dz}\Delta z$$ Substitute z, DH, and $\Delta q$. Solve for $\Delta z$.

ehild

3. Oct 12, 2014

Kyrios

so I get this:

$$DH = \frac{z(z+2) - z^2 q}{2} = 0.232$$
$$\frac{dq}{dz} = \frac{4DH - 2z}{z^3} = 66$$
$$\Delta q = \frac{dq}{dz} \Delta z$$
$$\Delta z = \frac{0.06}{66} = 9 \times 10^{-4}$$

The error seems really small..?

4. Oct 13, 2014

ehild

It is almost right. You made some small mistake when calculating DH. Check.

The error is small, but z=0.2, so its relative error is about 0.5 %.

ehild