A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is `60^(@)` . When he moves 50 m away from the bank, he finds the angle of elevation to be `30^(@)` . Calculate :
the height of the tree.

Answer

To solve the problem step by step, we will follow a structured approach using trigonometry.

### Step 1: Understand the Problem and Draw the Diagram
We have a tree on the opposite bank of a river, and a man standing on the bank observes the tree. The angle of elevation to the top of the tree is given as \(60^\circ\) when he is at point C. After moving 50 meters away to point D, the angle of elevation changes to \(30^\circ\).

Let:
- \(AB\) be the height of the tree (H).
- \(BC\) be the distance from the man to the base of the tree (X).
- \(BD\) be the distance from the man at point D to the base of the tree, which is \(X + 50\).

### Step 2: Set Up the First Equation Using Triangle ABC
From triangle \(ABC\):
\[ \tan(60^\circ) = \frac{AB}{BC} = \frac{H}{X} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we can write:
\[ \sqrt{3} = \frac{H}{X} \implies X = \frac{H}{\sqrt{3}} \]

### Step 3: Set Up the Second Equation Using Triangle ABD
From triangle \(ABD\):
\[ \tan(30^\circ) = \frac{AB}{BD} = \frac{H}{X + 50} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we can write:
\[ \frac{1}{\sqrt{3}} = \frac{H}{X + 50} \implies X + 50 = H\sqrt{3} \]

### Step 4: Substitute the Value of X
Now we have two equations:
1. \(X = \frac{H}{\sqrt{3}}\)
2. \(X + 50 = H\sqrt{3}\)

Substituting the value of \(X\) from the first equation into the second:
\[ \frac{H}{\sqrt{3}} + 50 = H\sqrt{3} \]

### Step 5: Solve for H
Rearranging the equation:
\[ 50 = H\sqrt{3} - \frac{H}{\sqrt{3}} \]
Factoring out \(H\):
\[ 50 = H\left(\sqrt{3} - \frac{1}{\sqrt{3}}\right) \]
To combine the terms in the parentheses:
\[ \sqrt{3} - \frac{1}{\sqrt{3}} = \frac{3 - 1}{\sqrt{3}} = \frac{2}{\sqrt{3}} \]
Thus, we have:
\[ 50 = H \cdot \frac{2}{\sqrt{3}} \]
Solving for \(H\):
\[ H = 50 \cdot \frac{\sqrt{3}}{2} = 25\sqrt{3} \]

### Step 6: Calculate the Height
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ H \approx 25 \cdot 1.732 \approx 43.3 \text{ meters} \]

### Final Answer
The height of the tree is approximately **43.3 meters**.

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