Question

You measure two quantities as $A=1.0m\pm 0.2m$, $B=2.0m\pm 0.2m$. We should report correct value for $\sqrt{AB}$ as

• A. $1.4m\pm 0.4m$
• B. $1.41m\pm 0.15m$
• C. $1.4m\pm 0.3m$
• D. $1.4m\pm 0.2m$

Answer 1

The correct option is D $1.4m\pm 0.2m$

Here, $A=1.0m\pm 0.2m$, $B=2.0m\pm 0.2m$
$AB=\left(1.0m\right)\left(2.0m\right)=2.0m^2$
$\sqrt{AB}=\sqrt{2.0m}=1.414m$

Rounding off to two significant figures, we get
$\sqrt{AB}=1.4m$
$\frac{\Delta AB}{AB}=\frac{1}{2}(\frac{\Delta A}{A}+\frac{\Delta B}{B})=\frac{1}{2}(\frac{0.2}{1.0}+\frac{0.2}{2.0})=\frac{0.3}{2}$
$\Delta \sqrt{AB}=\frac{0.3}{2}\times \sqrt{AB}=\frac{0.3}{2}\times 1.414=0.212m$

Rounding off to one significant figure, we get
$\Delta \sqrt{AB}=0.2m$
The correct value for $\sqrt{AB}$ is $1.4m\pm 0.2m$.