To determine the direction in which the man should hold the umbrella, we need to consider the velocities of both the rain and the man on the bicycle.

Given:
Velocity of rain (Vr) = 30 m/s (falling vertically)
Velocity of man (Vm) = 10 m/s (east to west)

We can break down the velocities into their respective components in order to analyze the situation more easily.

Components of Velocity:
The velocity of the rain can be split into two components: one in the east-west direction (Vr_x) and one in the vertical direction (Vr_y).

The velocity of the man can also be split into two components: one in the east-west direction (Vm_x) and one in the vertical direction (Vm_y).

Since the rain is falling vertically, the vertical component of its velocity (Vr_y) will be 30 m/s, while the east-west component (Vr_x) will be 0 m/s.

Since the man is riding the bicycle in the east-west direction, the vertical component of his velocity (Vm_y) will be 0 m/s, while the east-west component (Vm_x) will be 10 m/s.

Therefore, we have:
Vr_x = 0 m/s
Vr_y = 30 m/s
Vm_x = 10 m/s
Vm_y = 0 m/s

To find the direction in which the man should hold the umbrella, we need to consider the resultant velocity of the rain relative to the man.

Resultant Velocity:
The resultant velocity of the rain relative to the man can be found by subtracting the man's velocity components from the rain's velocity components.

Vr_x_relative = Vr_x - Vm_x = 0 m/s - 10 m/s = -10 m/s (westward)
Vr_y_relative = Vr_y - Vm_y = 30 m/s - 0 m/s = 30 m/s (upward)

Therefore, the resultant velocity of the rain relative to the man is in the direction of (-10 m/s, 30 m/s) or (-10 m/s westward, 30 m/s upward).

Direction of the Umbrella:
The direction in which the man should hold the umbrella is the direction opposite to the resultant velocity of the rain relative to the man.

Using trigonometry, we can find the angle that represents the direction of the resultant velocity.

tan(theta) = Vr_y_relative / Vr_x_relative
tan(theta) = (30 m/s) / (-10 m/s)
tan(theta) = -3

Therefore, the angle theta is equal to tan^(-1)(-3) or approximately -71.57 degrees.

Since the angle is negative, the direction in which the man should hold the umbrella is 71.57 degrees westward from the vertical or 71.57 degrees eastward from the west.

Thus, the correct answer is option D) tan^(-11/3).

V ( r)=30 ,v(man)=10 East to west put Right Angle triangle law ....... hope u can do this......