To a boy cycling at the rate of 4 km/h . eastward, the wind seems to blow directly from the north. But when he cycles at the rate of 7 km/h, it seems to blow from the north-east . The magnitude of the actual velocity of the wind is

Options

A. `5// sqrt""2`
B. `5 sqrt""2`
C. 5
D. `11//2`

Answer

To solve the problem, we need to analyze the situation using vector addition and relative velocity concepts.

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- The boy is cycling eastward at two different speeds: 4 km/h and 7 km/h.
- When he cycles at 4 km/h, the wind appears to come from the north.
- When he cycles at 7 km/h, the wind appears to come from the northeast.

2. **Setting Up the Vectors**:
- Let the velocity of the wind be \( \vec{W} \) with components \( W_x \) (east-west) and \( W_y \) (north-south).
- The boy's velocity \( \vec{B} \) is \( 4 \hat{i} \) km/h when he cycles at 4 km/h and \( 7 \hat{i} \) km/h when he cycles at 7 km/h.

3. **Analyzing the First Case (4 km/h)**:
- When the boy cycles at 4 km/h, the wind appears to come from the north. This means that the resultant velocity of the wind relative to the boy is directed southward.
- The relative velocity of the wind with respect to the boy can be expressed as:
\[ \vec{W} - \vec{B} = W_x - 4 \hat{i} + W_y \hat{j} \]
- Since it appears to come from the north, the eastward component must be zero:
\[ W_x - 4 = 0 \implies W_x = 4 \text{ km/h} \]

4. **Analyzing the Second Case (7 km/h)**:
- When the boy cycles at 7 km/h, the wind appears to come from the northeast. This means the resultant velocity of the wind relative to the boy has equal components in the east and north directions.
- The relative velocity can be expressed as:
\[ \vec{W} - \vec{B} = W_x - 7 \hat{i} + W_y \hat{j} \]
- Since it appears to come from the northeast, we have:
\[ W_x - 7 = W_y \]

5. **Substituting the Value of \( W_x \)**:
- From the first case, we found \( W_x = 4 \) km/h. Substituting this into the equation from the second case:
\[ 4 - 7 = W_y \implies W_y = -3 \text{ km/h} \]

6. **Finding the Magnitude of the Wind Velocity**:
- Now we can find the magnitude of the wind's velocity vector:
\[ |\vec{W}| = \sqrt{W_x^2 + W_y^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ km/h} \]

### Final Answer:
The magnitude of the actual velocity of the wind is **5 km/h**.