Rain is falling vertically at `10sqrt(3)` kph. A man is running at 10 kph. The angle with vertical at which he should hold his umbrella is

Options

A. `60^(@)`
B. `45^(@)`
C. `30^(@)`
D. `15^(@)`

Answer

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

### Step 1: Understand the scenario
- The rain is falling vertically downward with a speed of \( 10\sqrt{3} \) km/h.
- The man is running horizontally at a speed of 10 km/h.

### Step 2: Identify the components of motion
- The rain's velocity can be represented as a vertical vector \( V_r = 10\sqrt{3} \) km/h (downward).
- The man's velocity can be represented as a horizontal vector \( V_m = 10 \) km/h (to the right).

### Step 3: Determine the resultant velocity of the rain relative to the man
- To find the angle at which the man should hold his umbrella, we need to find the resultant velocity of the rain as observed by the man. This is done by combining the vertical and horizontal components of the rain's motion.
- The vertical component of the rain's velocity is \( 10\sqrt{3} \) km/h.
- The horizontal component (due to the man's running) is \( 10 \) km/h.

### Step 4: Use trigonometry to find the angle
- The angle \( \theta \) that the man should hold his umbrella with respect to the vertical can be found using the tangent function:
\[ \tan(\theta) = \frac{\text{horizontal velocity}}{\text{vertical velocity}} = \frac{V_m}{V_r} = \frac{10}{10\sqrt{3}} = \frac{1}{\sqrt{3}} \]

### Step 5: Calculate the angle
- From the tangent value, we can find \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \]
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), thus:
\[ \theta = 30^\circ \]

### Conclusion
The man should hold his umbrella at an angle of \( 30^\circ \) with respect to the vertical to avoid getting wet from the rain.

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