On walking 120 m towards a chimney in a horizonatal line through its base the angle of elevation of tip of the chimney changes from `30^@ " to " 45^@`. The height of the chimney is

Options

A. 120 m
B. 60(V3 - 1)m
C. 60(V3 + 1)m
D. None of these

Answer

To solve the problem, we will use the concept of right triangles and trigonometric ratios.

### Step-by-Step Solution:

1. **Understand the Situation**:
- Let the height of the chimney be \( h \).
- Initially, the observer is at point A, where the angle of elevation to the top of the chimney is \( 30^\circ \).
- After walking 120 m towards the chimney, the observer reaches point B, where the angle of elevation is \( 45^\circ \).

2. **Set Up the Right Triangles**:
- From point A (initial position), we can form a right triangle where:
- The opposite side is the height of the chimney \( h \).
- The adjacent side is the distance from point A to the base of the chimney, which we will denote as \( d \).
- From point B (after walking 120 m), the distance to the base of the chimney is \( d - 120 \).

3. **Apply Trigonometric Ratios**:
- For point A (angle of elevation \( 30^\circ \)):
\[ \tan(30^\circ) = \frac{h}{d} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{d} \implies h = \frac{d}{\sqrt{3}} \quad (1) \]

- For point B (angle of elevation \( 45^\circ \)):
\[ \tan(45^\circ) = \frac{h}{d - 120} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h}{d - 120} \implies h = d - 120 \quad (2) \]

4. **Equate the Two Expressions for \( h \)**:
- From equations (1) and (2):
\[ \frac{d}{\sqrt{3}} = d - 120 \]

5. **Solve for \( d \)**:
- Rearranging gives:
\[ d - \frac{d}{\sqrt{3}} = 120 \]
\[ d \left(1 - \frac{1}{\sqrt{3}}\right) = 120 \]
\[ d \left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right) = 120 \]
\[ d = \frac{120 \sqrt{3}}{\sqrt{3} - 1} \]

6. **Calculate \( h \)**:
- Substitute \( d \) back into either equation (1) or (2) to find \( h \). Using equation (2):
\[ h = d - 120 = \frac{120 \sqrt{3}}{\sqrt{3} - 1} - 120 \]

7. **Simplify to Find \( h \)**:
- To simplify:
\[ h = \frac{120 \sqrt{3} - 120(\sqrt{3} - 1)}{\sqrt{3} - 1} = \frac{120 \sqrt{3} - 120\sqrt{3} + 120}{\sqrt{3} - 1} = \frac{120}{\sqrt{3} - 1} \]
- Rationalizing the denominator:
\[ h = \frac{120(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{120(\sqrt{3} + 1)}{3 - 1} = 60(\sqrt{3} + 1) \]

### Final Answer:
The height of the chimney is \( 60(\sqrt{3} + 1) \) meters.