Math Problem Statement
Solution
To solve this problem, let’s break down the steps required based on the image you provided. We have a closed traverse with bearings and distances between stations, and we need to:
- Complete the Table: Determine the coordinates for stations , , and by applying the bearings and distances, using the provided coordinates of station as a reference point.
- Calculate Linear Misclosure: Determine how far off the computed ending point is from the starting point after completing the traverse.
- Calculate Traverse Accuracy: This involves determining the ratio of the linear misclosure to the total traverse length.
Step-by-Step Solution Outline
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Convert Bearings to Cartesian Changes: For each leg (e.g., ), use the bearing and distance to calculate the changes in (East) and (North) coordinates. We can use: where is the distance and is the bearing.
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Compute Coordinates: Using the coordinates of point (18.149 mN and 178.426 mE), apply the calculated and from the first step to find the coordinates for points , , and sequentially.
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Calculate Linear Misclosure: After finding the coordinates of , calculate the difference between and to check if they match. The discrepancy between the starting and calculated end point is the linear misclosure.
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Calculate Traverse Accuracy: The traverse accuracy is given by:
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Adjust Traverse Coordinates: Once we have the misclosure, we can distribute it across the traverse to correct each station’s coordinates and achieve a closed traverse.
Would you like me to proceed with detailed calculations, or do you have specific parts of the question you'd like to focus on?
Follow-Up Questions
- Would you like help in understanding how to convert bearings to Cartesian coordinates?
- Do you need clarification on how to calculate the linear misclosure?
- Are you interested in seeing the detailed calculation steps for each station's coordinates?
- Do you want to understand how traverse adjustment is applied to correct each point?
- Would you like an explanation on the significance of traverse accuracy and its implications?
Tip
In traverse computations, always double-check angle measurements and use consistent units (e.g., decimal degrees) for calculations to minimize errors.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Trigonometry
Surveying Techniques
Formulas
Δx = d * cos(θ)
Δy = d * sin(θ)
Traverse Accuracy = Linear Misclosure / Total Distance
Theorems
Pythagorean Theorem for calculating misclosure
Suitable Grade Level
Undergraduate level (Engineering/Surveying)
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