From a point 36 m above the surface of a lake , the angle of elevation of a bird is observed to be `30^(@)` and angle of depression of its image in the water of the lake is observed to be `60^(@)` . Find the actual of the bird above the surface of the lake .

Answer

To solve the problem step by step, we will use trigonometric concepts related to angles of elevation and depression.

### Step 1: Understand the Problem
We have a point 36 m above the lake's surface. From this point, we observe a bird at an angle of elevation of \(30^\circ\) and the angle of depression of its image in the water is \(60^\circ\). We need to find the actual height of the bird above the lake's surface.

### Step 2: Draw the Diagram
Draw a vertical line representing the height of the point above the lake (36 m). Mark the point as A. Below point A, draw the lake's surface and mark the point directly below A as B. Now, mark the position of the bird in the air as C and the image of the bird in the water as D.

### Step 3: Identify the Angles
- The angle of elevation from point A to the bird (C) is \(30^\circ\).
- The angle of depression from point A to the image of the bird (D) is \(60^\circ\).

### Step 4: Set Up the Right Triangles
1. **Triangle ACD** (for the angle of elevation):
- Let the height of the bird above the lake be \(h\).
- The total height from the lake to the bird is \(h + 36\) m.
- The horizontal distance from point A to point D is \(x\).

Using the tangent function:
\[ \tan(30^\circ) = \frac{h}{x} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies h = \frac{x}{\sqrt{3}} \quad (1) \]

2. **Triangle ABD** (for the angle of depression):
- The height from point A to point D is \(36\) m.
- The horizontal distance from point A to point C is also \(x\).

Using the tangent function:
\[ \tan(60^\circ) = \frac{36}{x} \]
Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{36}{x} \implies x = \frac{36}{\sqrt{3}} \quad (2) \]

### Step 5: Substitute to Find \(h\)
From equation (2), substitute \(x\) into equation (1):
\[ h = \frac{\frac{36}{\sqrt{3}}}{\sqrt{3}} = \frac{36}{3} = 12 \text{ m} \]

### Step 6: Calculate the Total Height of the Bird Above the Lake
The total height of the bird above the lake is:
\[ \text{Total height} = h + 36 = 12 + 36 = 48 \text{ m} \]

### Final Answer
The actual height of the bird above the surface of the lake is **48 meters**.

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