Find the height of the chimney when it is found that on walking towards it 50 meters in the horizontal line through its base, the angle of elevation of its top changes from `30^(@)` to `60^(@)`.

Answer

To find the height of the chimney, we can use the information given about the angles of elevation and the distances involved. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a chimney (AB) of height \( h \). Initially, we are at point D, where the angle of elevation to the top of the chimney is \( 30^\circ \). After walking 50 meters towards the chimney to point C, the angle of elevation changes to \( 60^\circ \).

### Step 2: Set Up the Triangles
1. **Triangle ADB** (where D is the initial position):
- The angle of elevation is \( 30^\circ \).
- Using the tangent function:
\[ \tan(30^\circ) = \frac{h}{AD} \]
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{AD} \implies AD = h \cdot \sqrt{3} \]

2. **Triangle ABC** (where C is the new position):
- The angle of elevation is \( 60^\circ \).
- Using the tangent function:
\[ \tan(60^\circ) = \frac{h}{BC} \]
- We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h}{BC} \implies BC = \frac{h}{\sqrt{3}} \]

### Step 3: Relate the Distances
From the problem, we know:
\[ AD = BC + 50 \]
Substituting the expressions we found:
\[ h \cdot \sqrt{3} = \frac{h}{\sqrt{3}} + 50 \]

### Step 4: Solve for \( h \)
1. Multiply through by \( \sqrt{3} \) to eliminate the fraction:
\[ h \cdot 3 = h + 50\sqrt{3} \]
2. Rearranging gives:
\[ 3h - h = 50\sqrt{3} \implies 2h = 50\sqrt{3} \]
3. Finally, divide by 2:
\[ h = \frac{50\sqrt{3}}{2} = 25\sqrt{3} \]

### Final Answer
The height of the chimney is \( 25\sqrt{3} \) meters.

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