A helicopter is in a stationary position at a certain height over the lake. At a point 200 m above the surface of the lake, the angle of elevation of the helicopter is `45^(@)`. At the same time, the angle of depression of its reflection in the lake is `75^(@)`. Calculate the height of the helicopter from the surface of the lake :

Options

A. `(200)/(sqrt3)`
B. `200sqrt3m`
C. 200 m
D. 150 m

Answer

To solve the problem, we will use trigonometric principles involving angles of elevation and depression.

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- Let the height of the helicopter from the surface of the lake be \( h \) meters.
- The observer is at a height of 200 m above the lake surface.
- The angle of elevation to the helicopter is \( 45^\circ \).
- The angle of depression to the helicopter's reflection in the lake is \( 75^\circ \).

2. **Setting Up the Diagram**:
- Draw a vertical line representing the height of the helicopter from the lake surface.
- Mark the observer's position 200 m above the lake.
- Draw a horizontal line from the observer to the point directly below the helicopter (on the lake surface).
- The angle of elevation to the helicopter is \( 45^\circ \) and the angle of depression to the reflection is \( 75^\circ \).

3. **Using the Angle of Elevation**:
- From the observer's position, the height of the helicopter can be represented using the tangent function:
\[ \tan(45^\circ) = \frac{h - 200}{d} \]
where \( d \) is the horizontal distance from the observer to the point directly below the helicopter.
- Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{h - 200}{d} \implies h - 200 = d \implies h = d + 200 \tag{1} \]

4. **Using the Angle of Depression**:
- The angle of depression to the reflection of the helicopter in the lake can also be represented using the tangent function:
\[ \tan(75^\circ) = \frac{200 + h}{d} \]
- Rearranging gives:
\[ d = \frac{200 + h}{\tan(75^\circ)} \tag{2} \]

5. **Substituting Equation (1) into Equation (2)**:
- Substitute \( h = d + 200 \) into \( d = \frac{200 + h}{\tan(75^\circ)} \):
\[ d = \frac{200 + (d + 200)}{\tan(75^\circ)} \]
\[ d = \frac{d + 400}{\tan(75^\circ)} \]
- Now, multiply both sides by \( \tan(75^\circ) \):
\[ d \tan(75^\circ) = d + 400 \]
\[ d \tan(75^\circ) - d = 400 \]
\[ d (\tan(75^\circ) - 1) = 400 \]
\[ d = \frac{400}{\tan(75^\circ) - 1} \]

6. **Calculating \( d \)**:
- Using \( \tan(75^\circ) \approx 3.732 \):
\[ d = \frac{400}{3.732 - 1} \approx \frac{400}{2.732} \approx 146.0 \text{ m} \]

7. **Finding the Height \( h \)**:
- Substitute \( d \) back into Equation (1):
\[ h = d + 200 \approx 146.0 + 200 \approx 346.0 \text{ m} \]

### Final Answer:
The height of the helicopter from the surface of the lake is approximately **346.0 meters**.