Aboy Walks 4m east and then 3m south .Find the displacement of the boy.

Options

A. 7
B. 1
C. 5
D. 12

Explanation

To solve the problem of finding the displacement of the boy who walks 4 meters east and then 3 meters south, we can follow these steps:

### Step 1: Understand the Movement
The boy first walks 4 meters to the east and then turns and walks 3 meters to the south. We can visualize this movement on a coordinate plane where:
- East corresponds to the positive x-direction.
- South corresponds to the negative y-direction.

### Step 2: Draw a Diagram
Draw a coordinate system:
- Mark the starting point (let's call it point A) at the origin (0,0).
- From point A, draw a line 4 meters to the right (east) to point B (4,0).
- From point B, draw a line 3 meters down (south) to point C (4,-3).

### Step 3: Identify Coordinates
- Point A (starting point): (0, 0)
- Point B (after walking east): (4, 0)
- Point C (after walking south): (4, -3)

### Step 4: Calculate the Displacement
Displacement is the straight-line distance from the starting point (A) to the final point (C). We can calculate this using the distance formula:

\[ \text{Displacement} = AC = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} \]

Substituting the coordinates:
- \(x_A = 0\), \(y_A = 0\)
- \(x_C = 4\), \(y_C = -3\)

\[ AC = \sqrt{(4 - 0)^2 + (-3 - 0)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ meters} \]

### Step 5: Determine the Direction of Displacement
To find the angle of displacement with respect to the east direction, we can use the tangent function:

\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AB} = \frac{3}{4} \]

Thus, the angle \(\theta\) can be found using the inverse tangent function:

\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]

### Final Answer
The displacement of the boy is 5 meters at an angle of \(\theta = \tan^{-1}\left(\frac{3}{4}\right)\) south of east.

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