The distance between two towers is 140 m while seeing from the top if the second tower, the angle of elevation of first tower is `30^(@)`.If the height of the second tower is 60 m, then find the height of the first tower.

Answer

To solve the problem step by step, we will use trigonometric concepts, particularly the tangent function, which relates the angles and sides of a right triangle.

### Step-by-step Solution:

1. **Understanding the Problem**:
- We have two towers, Tower A (height \( h \)) and Tower D (height \( 60 \, \text{m} \)).
- The distance between the two towers (horizontal distance) is \( 140 \, \text{m} \).
- The angle of elevation from the top of Tower D to the top of Tower A is \( 30^\circ \).

2. **Setting Up the Diagram**:
- Let \( A \) be the top of the first tower, \( B \) be the bottom of the first tower, \( C \) be the bottom of the second tower, and \( D \) be the top of the second tower.
- The height of Tower D (from \( C \) to \( D \)) is \( 60 \, \text{m} \).
- The horizontal distance \( CD \) is \( 140 \, \text{m} \).

3. **Identifying the Right Triangle**:
- We will focus on triangle \( ACD \) where:
- \( AC \) is the vertical distance from the top of Tower D to the top of Tower A, which is \( h - 60 \) (since \( h \) is the height of Tower A).
- \( CD \) is the horizontal distance between the two towers, which is \( 140 \, \text{m} \).

4. **Using the Tangent Function**:
- The tangent of the angle of elevation can be expressed as:
\[ \tan(30^\circ) = \frac{AC}{CD} \]
- Substituting the known values:
\[ \tan(30^\circ) = \frac{h - 60}{140} \]
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).

5. **Setting Up the Equation**:
- Substitute \( \tan(30^\circ) \) into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{h - 60}{140} \]

6. **Cross-Multiplying**:
- Cross-multiply to eliminate the fraction:
\[ 140 \cdot 1 = \sqrt{3}(h - 60) \]
- This simplifies to:
\[ 140 = \sqrt{3}(h - 60) \]

7. **Solving for \( h \)**:
- Divide both sides by \( \sqrt{3} \):
\[ h - 60 = \frac{140}{\sqrt{3}} \]
- Now, add \( 60 \) to both sides:
\[ h = 60 + \frac{140}{\sqrt{3}} \]

8. **Calculating the Height**:
- To find \( h \), calculate \( \frac{140}{\sqrt{3}} \):
\[ \frac{140}{\sqrt{3}} \approx 80.77 \quad (\text{using } \sqrt{3} \approx 1.732) \]
- Therefore:
\[ h \approx 60 + 80.77 \approx 140.77 \, \text{m} \]

9. **Final Answer**:
- The height of the first tower is approximately \( 140.77 \, \text{m} \).

### Final Result:
The height of the first tower is approximately \( 140.77 \, \text{m} \).