Worked exercise — Propagation of error in a product

Problem

You measure the base $b = (12{,}3 \pm 0{,}2)\ \text{cm}$ and the height $h = (7{,}5 \pm 0{,}1)\ \text{cm}$ of a rectangle. Calculate the area and its relative uncertainty.

Solution

Value of the area. $A = b\,h = 12{,}3 \cdot 7{,}5 = 92{,}25\ \text{cm}^2$

Relative errors on the two measurements. $\frac{\Delta b}{b} = \frac{0{,}2}{12{,}3} = 1{,}6\%\ ;\qquad \frac{\Delta h}{h} = \frac{0{,}1}{7{,}5} = 1{,}3\%$

Propagation for a product. For a product of independent quantities the relative errors add in quadrature: $\frac{\Delta A}{A} \approx \sqrt{\left(\frac{\Delta b}{b}\right)^2 + \left(\frac{\Delta h}{h}\right)^2} = \sqrt{1{,}6^2 + 1{,}3^2}\ \% \approx 2{,}1\%$

Absolute uncertainty. $\Delta A \approx 0{,}021 \cdot 92{,}25 \approx 1{,}9\ \text{cm}^2$

Rounding to the significant figure of the uncertainty: $\ev{A = (92 \pm 2)\ \text{cm}^2,\qquad \frac{\Delta A}{A} \approx 2\%}$

Links

Topics: Metodo sperimentale Concepts: Cifre significative e incertezza Skills: Propagazione degli errori · Uso delle cifre significative