Calculate focal length of a spherical mirror from the following observations.

Object distance, $u=(50.1\pm 0.5)$cm and image distance, $v=(20.1\pm 0.2)$ cm.

The correct option is D $f=(14.3\pm 0.4)$ cm
$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$
$f=\frac{uv}{u+v}=\frac{\left(50.1\right)\left(20.1\right)}{(50.1+20.1)}=14.3$ cm

Also, $\frac{\Delta f}{f}=\pm [\frac{\Delta u}{u}+\frac{\Delta u}{v}+\frac{\Delta u+\Delta v}{u+v}]$

$=\pm [\frac{0.5}{50.1}+\frac{0.2}{20.1}+\frac{0.5+0.2}{50.1+20.1}]$

$=\pm [0.00998+0.00995+0.00997]$

$=\pm \left(0.0299\right)$

$\therefore \Delta f=0.0299\times 14.3=0.428=0.4$ cm

$\therefore f=(14.3\pm 0.4)$ cm