Problem Statement
A survey line BAC crosses a river; A and C being the near and far banks respectively. A perpendicular AD, 40 metres long is set out at A. If the bearings of AD and DC are 38° 45′ and 278° 45′ respectively, find the width of the river.
Step-by-Step Solution
Key Information
- Line BAC crosses the river with A and C being the near and far banks
- Perpendicular AD = 40 metres
- Bearing of AD = 38° 45′
- Bearing of DC = 278° 45′
- Goal: Find the width of the river (distance AC)
Step 1: Understanding Bearings
The bearing of a line is defined as the horizontal angle between the north direction and the line measured in a clockwise direction.
We know:
Bearing of AD = 38° 45′
To find the bearing of DA (reverse direction), we add 180° to the bearing of AD:
Bearing of DA = 38° 45′ + 180° = 218° 45′
We also know:
Bearing of DC = 278° 45′
Step 2: Calculate the Angle CDA
To find angle CDA, we subtract the bearing of DA from the bearing of DC:
∠CDA = Bearing of DC − Bearing of DA
∠CDA = 278° 45′ − 218° 45′
∠CDA = 60°
Step 3: Calculate the Width of the River (AC)
From the triangle ACD, we can use trigonometry to find AC:
In triangle ACD, we know:
– Side AD = 40 metres
– Angle CDA = 60°
– Angle DAC = 90° (perpendicular)
Using the tangent relationship:
AC = AD × tan(∠CDA)
AC = 40 × tan(60°)
AC = 40 × √3
AC = 40 × 1.732
AC = 69.28 metres
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Bearings in Surveying: Bearings measure the horizontal angle clockwise from north to a line. Used to establish orientation of survey lines.
- Reverse Bearings: A fundamental concept in surveying where the bearing in the opposite direction differs by exactly 180°.
- Perpendicular Offsets: Setting perpendicular lines allows for measurement of otherwise inaccessible distances.
- Trigonometric Applications: Using right triangles and trigonometric ratios to calculate distances indirectly.
Real-World Applications:
- River Crossing Design: Essential for planning bridges and determining optimal crossing points.
- Utility Installation: Planning for pipelines or cables that need to cross water bodies.
- Land Boundary Surveys: Necessary when property lines cross natural features like rivers.
- Topographic Mapping: Creating accurate maps of terrain with water features.
- Environmental Studies: Measuring water body dimensions for watershed analysis and flood studies.
Why It Works:
This problem demonstrates how surveyors can measure distances that cannot be directly accessed, such as across a river. By establishing a perpendicular line of known length (AD = 40m) and measuring the bearings of lines AD and DC, we can use trigonometry to calculate the width of the river.
The key insight is using the difference in bearings to determine the angle CDA (60°). Once we know this angle and the length of AD, we can use the tangent function to find AC. Since AD is perpendicular to the river crossing line, the triangle ACD has a right angle at A, making this a right triangle calculation.
The formula AC = AD × tan(60°) gives us the river width as 69.28 meters. This method is particularly valuable because it allows measurements without physically crossing the river, which might be dangerous or impractical.
