A stationary person observes that rain is falling vertically down at `30 km h^-1`. A cyclist is moving up on an inclined plane making an angle `30^@` with horizontal at `10 km h^-1`. In which direction should the cyclist hold his umbrella to prevent himself from the rain ?

Options

A. At an angle `tan^(-1)(3sqrt(3))/(5)` with inclined plane
B. At an angle `tan^(-1) (3 sqrt(3))/(5)` with horizontal
C. At an `tan^(-1)(sqrt(3))/(7)` with inclined plane
D. At an angle `tan^(-1)(sqrt(3))/(7)` with vertical

Explanation

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

### Step 1: Understand the velocities involved
- The rain is falling vertically downwards at a speed of \( V_r = 30 \, \text{km/h} \).
- The cyclist is moving up an inclined plane at an angle of \( \theta = 30^\circ \) with the horizontal at a speed of \( V_c = 10 \, \text{km/h} \).

### Step 2: Break down the cyclist's velocity into components
- The cyclist's velocity can be resolved into two components:
- Horizontal component: \( V_{c,x} = V_c \cos(30^\circ) = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \, \text{km/h} \)
- Vertical component: \( V_{c,y} = V_c \sin(30^\circ) = 10 \cdot \frac{1}{2} = 5 \, \text{km/h} \)

### Step 3: Determine the relative velocity of rain with respect to the cyclist
- The velocity of the rain with respect to the cyclist can be calculated as:
\[ V_{r/c} = V_r - V_c \]
- Since the rain is falling downwards and the cyclist is moving upwards on the incline, we can express this as:
- Downward velocity of rain: \( V_r = 30 \, \text{km/h} \)
- Upward component of cyclist's velocity: \( V_{c,y} = 5 \, \text{km/h} \)

Thus, the effective downward velocity of the rain relative to the cyclist is:
\[ V_{r/c} = 30 - 5 = 25 \, \text{km/h} \]

### Step 4: Calculate the resultant velocity of rain relative to the cyclist
- The resultant velocity of the rain relative to the cyclist can be represented as a vector with:
- Vertical component: \( V_{r/c} = 25 \, \text{km/h} \) (downward)
- Horizontal component: \( V_{c,x} = 5\sqrt{3} \, \text{km/h} \) (upward)

### Step 5: Determine the angle at which the umbrella should be held
- We can find the angle \( \phi \) that the resultant velocity vector makes with the vertical using the tangent function:
\[ \tan(\phi) = \frac{\text{Horizontal Component}}{\text{Vertical Component}} = \frac{5\sqrt{3}}{25} = \frac{\sqrt{3}}{5} \]
- Therefore, the angle \( \phi \) can be calculated as:
\[ \phi = \tan^{-1}\left(\frac{\sqrt{3}}{5}\right) \]

### Step 6: Conclusion
- The cyclist should hold the umbrella at an angle \( \phi \) from the vertical towards the direction of the horizontal component of the rain's relative velocity.