Rain is falling vertically with velocity `10m/s` and a man is moving horizontally with velocity `6m/s` . The angle with which umbrella should be hold (with vertical) by a man to avoid getting wet is

Options

A. `tan^(-1)(5/3)`
B. `tan^(-1)(2/3)`
C. `tan^(-1)(3/5)`
D. `tan^(-1)(3/2)`

Answer

To solve the problem of determining the angle at which a man should hold his umbrella to avoid getting wet in the rain, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the velocities**:
- The rain is falling vertically with a velocity of \(10 \, \text{m/s}\) downward. We can represent this as a vector:
\[ \vec{V}_{\text{rain}} = -10 \hat{j} \, \text{m/s} \]
- The man is moving horizontally with a velocity of \(6 \, \text{m/s}\) to the right. We can represent this as a vector:
\[ \vec{V}_{\text{man}} = 6 \hat{i} \, \text{m/s} \]

2. **Determine the relative velocity of rain with respect to the man**:
- The velocity of rain with respect to the man is given by:
\[ \vec{V}_{\text{rain, man}} = \vec{V}_{\text{rain}} - \vec{V}_{\text{man}} = (-10 \hat{j}) - (6 \hat{i}) = -10 \hat{j} - 6 \hat{i} \]

3. **Visualize the vectors**:
- The vector \(-10 \hat{j}\) represents the downward velocity of the rain, and the vector \(-6 \hat{i}\) represents the horizontal velocity of the man. We can visualize this as a right triangle where:
- The vertical side (opposite) is \(10 \, \text{m/s}\)
- The horizontal side (adjacent) is \(6 \, \text{m/s}\)

4. **Calculate the angle**:
- To find the angle \(\theta\) at which the umbrella should be held, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{10} = \frac{3}{5} \]
- Therefore, we can find \(\theta\) by taking the arctangent:
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \]

5. **Final calculation**:
- Using a calculator, we can find:
\[ \theta \approx 30.96^\circ \]
- Thus, the angle with which the umbrella should be held with respect to the vertical is approximately \(30.96^\circ\).

### Final Answer:
The angle with which the umbrella should be held with the vertical to avoid getting wet is approximately \(30.96^\circ\).

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