To solve this problem and calculate the length and bearing of EA, we'll follow the steps typically used for traversing problems:

1. Extract Coordinates: Calculate the changes in coordinates (ΔX and ΔY) for each line using the formula: $\Delta X = L \cdot \cos(\theta), \quad \Delta Y = L \cdot \sin(\theta)$ where $L$ is the length of the line and $\theta$ is the bearing converted to decimal degrees.

2. Apply Closures:

   • The sum of the changes in $X$ ($\Sigma \Delta X$) and $Y$ ($\Sigma \Delta Y$) for the traverse should be zero in a closed loop. Since EA is omitted, use this condition to find its components.

3. Calculate EA:

   • Using the computed $\Delta X_{EA}$ and $\Delta Y_{EA}$, determine the length and bearing: $L = \sqrt{\Delta X^2 + \Delta Y^2}, \quad \text{Bearing} = \arctan\left(\frac{\Delta Y}{\Delta X}\right)$

Let me calculate the values step by pandas as pd import numpy as np

Data from the table

data = { "Line": ["AB", "BC", "CD", "DE", "EA"], "Length": [204, 220, 187, 192, None], # EA length is unknown "Bearing": ["87:30", "20:20", "280:30", "210:00", None] # EA bearing is unknown }

Convert bearings to decimal degrees

def bearing_to_decimal(bearing): if bearing is None: return None degrees, minutes = map(float, bearing.split(':')) return degrees + minutes / 60

Apply conversion to bearings

data["Bearing (Decimal)"] = [bearing_to_decimal(b) for b in data["Bearing"]]

Create a DataFrame

df = pd.DataFrame(data)

Calculate ΔX and ΔY for each line

df["Delta_X"] = df["Length"] * np.cos(np.radians(df["Bearing (Decimal)"])) df["Delta_Y"] = df["Length"] * np.sin(np.radians(df["Bearing (Decimal)"]))

Compute the closures for ΔX and ΔY (excluding EA since it's unknown)

closure_x = -df["Delta_X"].sum(skipna=True) closure_y = -df["Delta_Y"].sum(skipna=True)

Calculate the length and bearing for EA

length_ea = np.sqrt(closure_x2 + closure_y2) bearing_ea = np.degrees(np.arctan2(closure_y, closure_x))

Update the DataFrame with the computed EA values

df.loc[df["Line"] == "EA", "Delta_X"] = closure_x df.loc[df["Line"] == "EA", "Delta_Y"] = closure_y df.loc[df["Line"] == "EA", "Length"] = length_ea df.loc[df["Line"] == "EA", "Bearing (Decimal)"] = bearing_ea

Show the final data

import ace_tools as tools; tools.display_dataframe_to_user(name="Traverse Data with EA Calculation", dataframe=df)

The calculated length and bearing for the line EA are as follows:

• Length of EA: 82.99 meters
• Bearing of EA: -179.74° (approximately)

I have presented the full traversal data in a table for your review. Let me know if you need detailed explanations or further questions.

Questions to Expand:

1. What is the importance of bearings in traverse computations?
2. How can traverse misclosure be adjusted in practical surveys?
3. What tools can be used to directly measure bearings in the field?
4. How does the bearing quadrant affect the calculation of ΔX and ΔY?
5. What are other methods to compute the traverse closure?

Tip:

Always check for calculation errors or misclosure in field data, as these can affect the accuracy of the traverse