A body lost in a jungle finds a note. In the note was written the following things.
Displacements
1. 300 m `53^(@)` South of East .
2. 400 m `37^(@)` North of East
3. 500 m North
4. `500sqrt(2) m ` North- West
5. 500 m South
He starts walking at constant speed 2m/s following these displacements in the given order.
How far and in which direction is he from teh starting point after 10 minutes ?

Options

A. `500sqrt(2)` m due North
B. 1200 m due North- East
C. `500sqrt(2)` m due North- East
D. 900 m due `37^(@)` North of East

Answer

To solve the problem step by step, we need to calculate the resultant displacement of the body after following the given displacements in the specified order.

### Step 1: Break down each displacement into components

1. **Displacement 1: 300 m South of East at 53°**
- East component: \( 300 \cos(53°) \)
- South component: \( 300 \sin(53°) \)

2. **Displacement 2: 400 m North of East at 37°**
- East component: \( 400 \cos(37°) \)
- North component: \( 400 \sin(37°) \)

3. **Displacement 3: 500 m North**
- North component: 500 m

4. **Displacement 4: \( 500\sqrt{2} \) m Northwest (which is 45° from both North and West)**
- North component: \( 500\sqrt{2} \cos(45°) \)
- West component: \( 500\sqrt{2} \sin(45°) \)

5. **Displacement 5: 500 m South**
- South component: 500 m

### Step 2: Calculate the components for each displacement

1. **For 300 m South of East at 53°:**
- East: \( 300 \cos(53°) = 300 \times 0.6018 \approx 180.54 \, m \)
- South: \( 300 \sin(53°) = 300 \times 0.7986 \approx 239.58 \, m \)

2. **For 400 m North of East at 37°:**
- East: \( 400 \cos(37°) = 400 \times 0.7986 \approx 319.44 \, m \)
- North: \( 400 \sin(37°) = 400 \times 0.6018 \approx 240.72 \, m \)

3. **For 500 m North:**
- North: 500 m

4. **For \( 500\sqrt{2} \) m Northwest:**
- North: \( 500\sqrt{2} \cos(45°) = 500\sqrt{2} \times 0.7071 \approx 353.55 \, m \)
- West: \( 500\sqrt{2} \sin(45°) = 500\sqrt{2} \times 0.7071 \approx 353.55 \, m \)

5. **For 500 m South:**
- South: 500 m

### Step 3: Sum up all the components

- **Total East Component:**
\[ E = 180.54 + 319.44 = 499.98 \, m \]

- **Total North Component:**
\[ N = 240.72 + 500 + 353.55 - 239.58 - 500 = 354.69 \, m \]

### Step 4: Calculate the resultant displacement

Using Pythagorean theorem:
\[ R = \sqrt{E^2 + N^2} = \sqrt{(499.98)^2 + (354.69)^2} \]

Calculating:
\[ R = \sqrt{249980 + 125800} \approx \sqrt{375780} \approx 613.09 \, m \]

### Step 5: Determine the direction

Using the tangent function:
\[ \tan(\theta) = \frac{N}{E} = \frac{354.69}{499.98} \]
Calculating:
\[ \theta = \tan^{-1}\left(\frac{354.69}{499.98}\right) \approx 36.87° \]

### Final Result

The body is approximately **613.09 m** at an angle of **36.87° North of East** from the starting point after 10 minutes.