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The horizontal distance between two towers is 140 meter The angle of elevation of the top of the first tower,when seen from the top of the second tower is 30° If the height of the second tower is 60m find the height of the tower?
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The horizontal distance between two towers is 140 meter The angle of e...
To find the height of the first tower given the height of the second tower and the angle of elevation, we can use trigonometric principles.
Given Information
- Horizontal distance between the two towers: 140 meters
- Angle of elevation from the top of the second tower to the top of the first tower: 30°
- Height of the second tower: 60 meters
Diagram Representation
1. **Visualize the Towers**: Imagine two vertical towers. The first tower is taller, and the second tower is 60 meters tall.
2. **Positioning**: The distance between the bases of the towers is 140 meters.
Calculating the Height of the First Tower
- **Identify the Right Triangle**: From the top of the second tower, draw a line to the top of the first tower, forming a right triangle.
- **Use Tangent Function**: In the right triangle, the tangent of the angle of elevation (30°) is equal to the opposite side (height difference between the two towers) divided by the adjacent side (horizontal distance).
\[
\tan(30°) = \frac{h_1 - h_2}{d}
\]
Where:
- \( h_1 \) = height of the first tower
- \( h_2 = 60 \, m \) (height of the second tower)
- \( d = 140 \, m \) (horizontal distance)
- **Using Tangent Value**: The value of \( \tan(30°) = \frac{1}{\sqrt{3}} \).
Thus, the equation becomes:
\[
\frac{1}{\sqrt{3}} = \frac{h_1 - 60}{140}
\]
Solving for Height
1. **Cross-Multiply**:
\[
h_1 - 60 = \frac{140}{\sqrt{3}}
\]
2. **Add 60**:
\[
h_1 = 60 + \frac{140}{\sqrt{3}}
\]
3. **Calculate**:
\[
h_1 \approx 60 + 80.0 \approx 140 \, \text{m}
\]
Final Result
- The height of the first tower is approximately **140 meters**.
Given Information
- Horizontal distance between the two towers: 140 meters
- Angle of elevation from the top of the second tower to the top of the first tower: 30°
- Height of the second tower: 60 meters
Diagram Representation
1. **Visualize the Towers**: Imagine two vertical towers. The first tower is taller, and the second tower is 60 meters tall.
2. **Positioning**: The distance between the bases of the towers is 140 meters.
Calculating the Height of the First Tower
- **Identify the Right Triangle**: From the top of the second tower, draw a line to the top of the first tower, forming a right triangle.
- **Use Tangent Function**: In the right triangle, the tangent of the angle of elevation (30°) is equal to the opposite side (height difference between the two towers) divided by the adjacent side (horizontal distance).
\[
\tan(30°) = \frac{h_1 - h_2}{d}
\]
Where:
- \( h_1 \) = height of the first tower
- \( h_2 = 60 \, m \) (height of the second tower)
- \( d = 140 \, m \) (horizontal distance)
- **Using Tangent Value**: The value of \( \tan(30°) = \frac{1}{\sqrt{3}} \).
Thus, the equation becomes:
\[
\frac{1}{\sqrt{3}} = \frac{h_1 - 60}{140}
\]
Solving for Height
1. **Cross-Multiply**:
\[
h_1 - 60 = \frac{140}{\sqrt{3}}
\]
2. **Add 60**:
\[
h_1 = 60 + \frac{140}{\sqrt{3}}
\]
3. **Calculate**:
\[
h_1 \approx 60 + 80.0 \approx 140 \, \text{m}
\]
Final Result
- The height of the first tower is approximately **140 meters**.
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The horizontal distance between two towers is 140 meter The angle of elevation of the top of the first tower,when seen from the top of the second tower is 30° If the height of the second tower is 60m find the height of the tower?
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The horizontal distance between two towers is 140 meter The angle of elevation of the top of the first tower,when seen from the top of the second tower is 30° If the height of the second tower is 60m find the height of the tower? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The horizontal distance between two towers is 140 meter The angle of elevation of the top of the first tower,when seen from the top of the second tower is 30° If the height of the second tower is 60m find the height of the tower? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The horizontal distance between two towers is 140 meter The angle of elevation of the top of the first tower,when seen from the top of the second tower is 30° If the height of the second tower is 60m find the height of the tower?.
The horizontal distance between two towers is 140 meter The angle of elevation of the top of the first tower,when seen from the top of the second tower is 30° If the height of the second tower is 60m find the height of the tower? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The horizontal distance between two towers is 140 meter The angle of elevation of the top of the first tower,when seen from the top of the second tower is 30° If the height of the second tower is 60m find the height of the tower? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The horizontal distance between two towers is 140 meter The angle of elevation of the top of the first tower,when seen from the top of the second tower is 30° If the height of the second tower is 60m find the height of the tower?.
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