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The angle of elevation of the top of a light house 60m high,from two points on the ground on its opposite sides are 45°and 60° . what is the distance between these two points?
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The angle of elevation of the top of a light house 60m high,from two p...
To solve the problem of finding the distance between two points on the ground from which the angle of elevation to the top of a 60m high lighthouse is 45° and 60°, we can use trigonometric principles.
Understanding the Scenario
- We have a lighthouse that is 60 meters tall.
- The angles of elevation from two points on the opposite sides are 45° and 60°.
Using Trigonometry
1. Point A (Angle = 45°)
- From point A, using the tangent function:
\[
\tan(45°) = \frac{\text{Height of lighthouse}}{\text{Distance from point A to lighthouse}} \implies 1 = \frac{60}{d_1} \implies d_1 = 60m
\]
2. Point B (Angle = 60°)
- From point B, using the tangent function:
\[
\tan(60°) = \frac{\text{Height of lighthouse}}{\text{Distance from point B to lighthouse}} \implies \sqrt{3} = \frac{60}{d_2} \implies d_2 = \frac{60}{\sqrt{3}} = 20\sqrt{3} \approx 34.64m
\]
Calculating the Distance Between Points A and B
- The total distance between the two points is the sum of \(d_1\) and \(d_2\):
\[
\text{Distance} = d_1 + d_2 = 60 + 20\sqrt{3} \approx 60 + 34.64 = 94.64m
\]
Final Result
- The distance between the two points on the ground is approximately 94.64 meters.
This approach effectively applies the properties of right triangles and trigonometric ratios to determine the required distance.
Understanding the Scenario
- We have a lighthouse that is 60 meters tall.
- The angles of elevation from two points on the opposite sides are 45° and 60°.
Using Trigonometry
1. Point A (Angle = 45°)
- From point A, using the tangent function:
\[
\tan(45°) = \frac{\text{Height of lighthouse}}{\text{Distance from point A to lighthouse}} \implies 1 = \frac{60}{d_1} \implies d_1 = 60m
\]
2. Point B (Angle = 60°)
- From point B, using the tangent function:
\[
\tan(60°) = \frac{\text{Height of lighthouse}}{\text{Distance from point B to lighthouse}} \implies \sqrt{3} = \frac{60}{d_2} \implies d_2 = \frac{60}{\sqrt{3}} = 20\sqrt{3} \approx 34.64m
\]
Calculating the Distance Between Points A and B
- The total distance between the two points is the sum of \(d_1\) and \(d_2\):
\[
\text{Distance} = d_1 + d_2 = 60 + 20\sqrt{3} \approx 60 + 34.64 = 94.64m
\]
Final Result
- The distance between the two points on the ground is approximately 94.64 meters.
This approach effectively applies the properties of right triangles and trigonometric ratios to determine the required distance.
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The angle of elevation of the top of a light house 60m high,from two points on the ground on its opposite sides are 45°and 60° . what is the distance between these two points?
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The angle of elevation of the top of a light house 60m high,from two points on the ground on its opposite sides are 45°and 60° . what is the distance between these two points? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The angle of elevation of the top of a light house 60m high,from two points on the ground on its opposite sides are 45°and 60° . what is the distance between these two points? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The angle of elevation of the top of a light house 60m high,from two points on the ground on its opposite sides are 45°and 60° . what is the distance between these two points?.
The angle of elevation of the top of a light house 60m high,from two points on the ground on its opposite sides are 45°and 60° . what is the distance between these two points? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The angle of elevation of the top of a light house 60m high,from two points on the ground on its opposite sides are 45°and 60° . what is the distance between these two points? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The angle of elevation of the top of a light house 60m high,from two points on the ground on its opposite sides are 45°and 60° . what is the distance between these two points?.
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