Rain is falling vertically with a speed of 30 m/s. A woman rides a bicycle with a speed of 10 m/s in the north to south direction. What is the direction in which she should hold her umbrella?

Velocity of rain falling vertically downward

$v_r=30m/s$

Velocity of woman riding a bicycle

$v_w=10m/s\left(northtosouth\right)$

To protect herself from rain, the woman should hold her umbrella in the direction of relative velocity of the rain with respect to the woman, i.e.,  $v_{rw}$

The relative velocity of rain with respect to the woman, i.e., $v_{rw}=v_r-v_w$

$\therefore \left|v_{rw}\right|=\sqrt{{\left(30\right)}^2+{\left(10\right)}^2}$

$=\sqrt{900+100}=\sqrt{1000}m/s$

$=10\sqrt{10}m/s$

If $v_{rw}$ makes an angle $\alpha$ with the vertical, then

$\tan \alpha =\frac{v_w}{r_r}=\frac{10}{30}$

$=\frac{1}{3}=0.3333$

or $\alpha ={18}^{0}26\text{'}$

Hence, woman should hold her umbrella at an angle of ${18}^{0}26\text{'}$ with the vertical towards south.