Rain is falling vertically with a speed of `20ms^(-1)`., A person is running in the rain with a velocity of `5 ms^(-1)` and a wind is also blowing with a speed of `15 ms^(-1)` (both from the west) The angle with the vertical at which the person should hold his umbrella so that he may not get drenched is:

Answer

To solve the problem, we need to find the angle at which a person should hold his umbrella to avoid getting drenched while running in the rain. We will analyze the velocities involved and use trigonometry to find the required angle.

### Step-by-Step Solution:

1. **Identify the Velocities:**
- The rain is falling vertically with a speed of \( V_r = 20 \, \text{m/s} \) (downward).
- The person is running horizontally with a speed of \( V_m = 5 \, \text{m/s} \) (from west to east).
- The wind is blowing horizontally with a speed of \( V_w = 15 \, \text{m/s} \) (from west to east).

2. **Determine the Effective Velocity of Rain:**
- The effective velocity of rain with respect to the ground can be represented as a vector:
\[ \vec{V}_{\text{rain}} = 0 \, \hat{i} - 20 \, \hat{j} = -20 \, \hat{j} \]
- The effective velocity of wind can be represented as:
\[ \vec{V}_{\text{wind}} = 15 \, \hat{i} + 0 \, \hat{j} = 15 \, \hat{i} \]

3. **Calculate the Resultant Velocity of Rain with Respect to the Person:**
- The velocity of rain with respect to the person can be calculated by subtracting the velocity of the person from the velocity of rain:
\[ \vec{V}_{\text{rain, person}} = \vec{V}_{\text{rain}} - \vec{V}_{\text{man}} = (-20 \, \hat{j}) - (5 \, \hat{i}) = -20 \, \hat{j} - 5 \, \hat{i} \]
- This gives us:
\[ \vec{V}_{\text{rain, person}} = -5 \, \hat{i} - 20 \, \hat{j} \]

4. **Calculate the Magnitude of the Resultant Velocity:**
- The magnitude of the resultant velocity can be calculated using the Pythagorean theorem:
\[ V_{\text{resultant}} = \sqrt{(-5)^2 + (-20)^2} = \sqrt{25 + 400} = \sqrt{425} = 5\sqrt{17} \, \text{m/s} \]

5. **Determine the Angle with the Vertical:**
- To find the angle \( \theta \) that the umbrella should be held, we use the tangent function:
\[ \tan(\theta) = \frac{\text{horizontal component}}{\text{vertical component}} = \frac{5}{20} = \frac{1}{4} \]
- Therefore, the angle \( \theta \) can be calculated as:
\[ \theta = \tan^{-1}\left(\frac{1}{4}\right) \]

6. **Final Calculation:**
- Using a calculator, we find:
\[ \theta \approx 14.04^\circ \]

### Conclusion:
The angle with the vertical at which the person should hold his umbrella so that he may not get drenched is approximately \( 14.04^\circ \).