She should hold the umbrella a little inclined towards the west from the vertical.

The drops which are falling a little ahead of the current position of the woman, will fall on her, because she moves in that direction.

relative velocity of the rain drops wrt cyclist    
= velocity of woman - velocity of rain

the magnitude of relative velocity the rain drops  wrt  woman     = √(12² + 35²) = 37 m/s

direction of relative velocity of the rain drops =  tan⁻¹ (12/35)  =  18.92  deg. from the vertical.

So the cyclist should hold the umbrella at 18.92 deg to the vertical in the forward direction.

When a person is cycling in the rain, it is necessary for them to hold their umbrella in a specific direction to prevent themselves from getting wet. In this problem, we are given the speed of the rain and the speed of the woman cycling. We need to determine the direction in which the woman should hold her umbrella to avoid getting wet.


The rain is falling vertically with a speed of 35 m s–1. This means that the rain is falling straight down from the sky. The woman is cycling with a speed of 12 m s–1 in an east to west direction. This means that she is moving from the east to the west. We need to determine the direction in which she should hold her umbrella to avoid getting wet.


To determine the direction in which the woman should hold her umbrella, we need to consider the relative motion of the woman and the rain. Since the rain is falling vertically, it will hit the woman at an angle when she is cycling. This angle will depend on the speed of the woman and the speed of the rain. If the woman is cycling directly towards the rain, then the rain will hit her face and upper body. If she is cycling directly away from the rain, then the rain will hit her back. Therefore, she needs to hold her umbrella at an angle to prevent herself from getting wet.


Since the woman is cycling from east to west, the rain will hit her at an angle from the north-east direction. Therefore, she should hold her umbrella in a north-east direction to prevent herself from getting wet. The angle at which she should hold her umbrella will depend on the speed of the rain and the speed of the woman. If the rain is falling faster than the woman is cycling, then she will need to hold her umbrella at a greater angle to prevent herself from getting wet.


In conclusion, when a person is cycling in the rain, they should hold their umbrella in the direction that is opposite to the direction from which the rain is falling. In this case, since the woman is cycling from east to west, she should hold her umbrella in the north-east direction to prevent herself from getting wet.