Math Problem Statement
- A closed traverse has the following data:
COURSE
BEARING
DISTANCE
A-B
N 9.27° E
58.7 m
B-C
S 88.43° E
27.3 m
C-D
N 86.73° E
35.2 m
D-E
S5.3° E
35.0 m
E-A
Table 10
Compute the bearing and the distance of line E-A.
Solution
To solve for the bearing and distance of line E-A in a closed traverse, we need to first understand the approach using a mathematical method called traverse computation. Here's the outline of the steps:
- Convert bearings to azimuths for each course.
- Resolve each distance into its North-South and East-West components using trigonometry (sine and cosine functions).
- Sum up the North-South and East-West components for all courses.
- Calculate the total misclosure in both North-South and East-West directions.
- Compute the distance and bearing of line E-A using the misclosure values.
Step 1: Convert Bearings to Azimuths
Azimuths are measured from the North in a clockwise direction:
- A-B:
- B-C:
- C-D:
- D-E:
Step 2: Resolve Each Distance into North-South and East-West Components
We now resolve each course's distance into its Northing (N-S) and Easting (E-W) components using sine and cosine:
-
A-B:
- Northing:
- Easting:
-
B-C:
- Northing:
- Easting:
-
C-D:
- Northing:
- Easting:
-
D-E:
- Northing:
- Easting:
Let's compute these values.
Step 3: Sum Northing and Easting Components
We add up all the Northings and Eastings to determine the total net change in North and East directions.
Step 4: Calculate Misclosure and Compute Line E-A
The misclosure gives us the net discrepancy in the North-South and East-West directions, and from this, we can find the bearing and distance of line E-A.
Let me calculate these values step-by-step.It seems like I can’t do more advanced data analysis right now. Please try again later.
If you need further clarification or help with any specific part of the solution, feel free to ask!
Here are five questions to expand your understanding:
- How are azimuths related to bearings in traverse computations?
- What is the purpose of resolving distances into Northing and Easting components?
- How can we handle misclosure errors in a closed traverse?
- What role does trigonometry play in surveying and traverse computations?
- How do you compute the distance and bearing from coordinate differences?
Tip: Always double-check the quadrant for bearings when converting from azimuths to avoid mistakes.
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Math Problem Analysis
Mathematical Concepts
Traverse Computation
Trigonometry
Azimuth Conversion
Bearing Calculation
Formulas
Northing = Distance × cos(Bearing Angle)
Easting = Distance × sin(Bearing Angle)
Azimuth Conversion: Adjust bearing based on the quadrant
Distance = √(Northing^2 + Easting^2)
Bearing = tan⁻¹(Easting / Northing)
Theorems
Pythagorean Theorem for distance
Trigonometric Functions (Sine, Cosine, and Tangent) in plane surveying
Suitable Grade Level
Undergraduate Engineering or Advanced High School
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