Rain is falling vertically with a velocity of `25 ms^-1`. A woman rides a bicycle with a speed of `10 ms^-1` in the north to south direction. What is the direction (angle with vertical) in which she should hold her umbrella to safe herself from rain ?

Options

A. `tan^-1 (0.4)`
B. `tan^-1 (1)`
C. `tan^-1 (sqrt(3))`
D. `tan^-1 (2.6)`

Explanation

To solve the problem of determining the angle at which the woman should hold her umbrella to shield herself from the rain, we can follow these steps:

### Step 1: Understand the velocities involved
- The rain is falling vertically downwards with a velocity \( V_r = 25 \, \text{m/s} \).
- The woman is riding her bicycle from north to south with a velocity \( V_w = 10 \, \text{m/s} \).

### Step 2: Visualize the situation
- Draw a diagram where the vertical line represents the direction of the rain falling downwards.
- The horizontal line represents the direction of the woman riding her bicycle.

### Step 3: Determine the resultant velocity of the rain with respect to the woman
- The effective velocity of the rain as perceived by the woman is the vector sum of the rain's velocity and the negative of the woman's velocity (since she is moving in the opposite direction).
- This can be expressed as:
\[ V_{rw} = V_r \text{ (downward)} + (-V_w) \text{ (horizontal)} \]

### Step 4: Use trigonometry to find the angle
- The angle \( \theta \) that the umbrella should make with the vertical can be found using the tangent function:
\[ \tan(\theta) = \frac{\text{horizontal velocity}}{\text{vertical velocity}} = \frac{V_w}{V_r} \]
- Substituting the values:
\[ \tan(\theta) = \frac{10 \, \text{m/s}}{25 \, \text{m/s}} = \frac{2}{5} = 0.4 \]

### Step 5: Calculate the angle \( \theta \)
- To find the angle \( \theta \), take the arctangent (inverse tangent) of 0.4:
\[ \theta = \tan^{-1}(0.4) \]

### Step 6: Final Calculation
- Using a calculator, we find:
\[ \theta \approx 21.8^\circ \]

### Conclusion
The woman should hold her umbrella at an angle of approximately \( 21.8^\circ \) with respect to the vertical to protect herself from the rain.

---