# Using differentials to determine the maximum error

Imagine this triangle

and the distance between city A and C.

Unfortunately, the clouds block the measurements of our telemetric devices. However, we once measured the distance between city A and B as $7$ km and the device we applied at that time had a measurement error of $0.1$ km.

We now measure the distance between city B and C with a second device bearing measurement error of $0.3$ km. We find that distance to be $4$ km.

Calculating the distance between A and C

Using the pythagorean theorem I calculate

$$b=\sqrt{7^2-4^2}$$ $$b=\sqrt{33}$$

What is the maximum error occuring with my approach?

I know I'm supposed to use differentials to calculate the maxiumum error here. But I'm unable to come up with an equation at the moment. Help is very much appreciated.

$$\frac{\Delta y}{y}=\frac{1}{2}.\frac{\Delta x_1}{x_2}+\frac{1}{2}.\frac{\Delta x_2}{x_2}$$
You already have calculated $y$. You have $x_1 =7$ with error of $\Delta x_1=0.1$ . Same with $x_2=4$ with error of $\Delta x_2=0.3$.(please assum units in $km$)
Simply calculate $\Delta y$
• Little typo $x_2$ instead of $x_1$ in the third fraction – Jean Marie May 2 '17 at 14:36
• Thanks! This helps. One further question: How do I compare $\Delta y$ with the error that would have occurred during a measurement without clouds based on the second device? – Phaneron May 2 '17 at 14:36