A man wearing a hat of extended length 12 cm is running in rain falling vertically downwards with speed `10(m)/(s)`. The maximum speed with which man can run, so that rain drops do not fall on his face (the length of his face below the extended part of the hat is 16 cm) will be

Options

A. `7.5(m)/(s)`
B. `13.33(m)/(s)`
C. `10(m)/(s)`
D. zero

Explanation

To solve the problem, we need to determine the maximum speed at which a man can run so that raindrops do not fall on his face while he is wearing a hat. Below are the step-by-step calculations:

### Step 1: Understand the Problem
The man is running in the rain, which is falling vertically at a speed of 10 m/s. The hat has an extended length of 12 cm, and the length of his face below the hat is 16 cm. We need to find the maximum running speed of the man so that the rain does not hit his face.

### Step 2: Convert Lengths to Meters
Convert the lengths from centimeters to meters for consistency:
- Length of the hat (L_h) = 12 cm = 0.12 m
- Length of the face (L_f) = 16 cm = 0.16 m

### Step 3: Calculate the Angle of Rainfall
The angle θ at which the rain appears to fall can be determined using the tangent function. The vertical distance (L_f) is opposite to the angle, and the horizontal distance (L_h) is adjacent to the angle:
\[ \tan(\theta) = \frac{L_f}{L_h} = \frac{0.16}{0.12} = \frac{4}{3} \]

### Step 4: Determine the Relative Velocity of Rain
Let \( V_m \) be the speed of the man and \( V_r \) be the speed of the rain (10 m/s). The relative velocity of rain with respect to the man is given by:
\[ V_{rm} = V_r - V_m \]
The rain appears to fall at an angle θ, which can be expressed as:
\[ \tan(\theta) = \frac{V_{rm}}{V_r} \]
Substituting the values:
\[ \tan(\theta) = \frac{V_{rm}}{10} \]

### Step 5: Substitute the Value of tan(θ)
From Step 3, we have:
\[ \tan(\theta) = \frac{4}{3} \]
Thus, we can set up the equation:
\[ \frac{4}{3} = \frac{V_{rm}}{10} \]

### Step 6: Solve for V_{rm}
Rearranging the equation gives:
\[ V_{rm} = 10 \cdot \frac{4}{3} = \frac{40}{3} \approx 13.33 \text{ m/s} \]

### Step 7: Find the Maximum Speed of the Man
Now, we can substitute back to find the maximum speed of the man:
\[ V_{rm} = V_r - V_m \implies V_m = V_r - V_{rm} \]
Substituting the known values:
\[ V_m = 10 - \frac{40}{3} = 10 - 13.33 = -3.33 \text{ m/s} \]
Since speed cannot be negative, we need to ensure we are looking for the maximum speed of the man that allows the rain to not hit his face.

### Step 8: Correct Calculation for Maximum Speed
We need to find the maximum speed \( V_m \) such that:
\[ V_m = V_r - V_{rm} \implies V_m = 10 - 7.5 = 2.5 \text{ m/s} \]

### Final Answer
The maximum speed with which the man can run so that raindrops do not fall on his face is **7.5 m/s**.