Math Problem Statement

How it was solve individually

How it was solved individually?

Solution

The table you provided seems to be a representation of a survey or a polygonal adjustment. To solve this individually, here's an explanation of how it may have been approached:

  1. Data Given: The table lists various stations (A, B, C, D) with lines and their corresponding measured distances and included angles. The angles between each pair of adjacent lines are included as well.

  2. Total Computed Interior Angle: The total computed interior angle (326.90°) is calculated by adding all the included angles. Each angle is between two lines coming from the same station.

  3. Theoretical Sum of Interior Angles: The theoretical sum of interior angles for any polygon can be calculated using the formula: Sum of Interior Angles=(n2)×180\text{Sum of Interior Angles} = (n - 2) \times 180^\circ where nn is the number of sides (stations). In your case, the theoretical sum is given as 360°, which suggests that the polygon has 4 sides (because (42)×180=360(4-2) \times 180 = 360^\circ).

  4. Relative Precision: The relative precision of the survey is given as 1/124. This is typically calculated by comparing the difference between the theoretical sum and the computed sum of angles with the total number of sides or stations.

    Relative precision is usually calculated with: Relative Precision=Error in AngleTotal Measured Angles=Theoretical Sum of AnglesTotal Computed Interior AngleTotal Computed Interior Angle\text{Relative Precision} = \frac{\text{Error in Angle}}{\text{Total Measured Angles}} = \frac{\text{Theoretical Sum of Angles} - \text{Total Computed Interior Angle}}{\text{Total Computed Interior Angle}}

    In this case: Relative Precision=360326.90360=33.103601/124\text{Relative Precision} = \frac{360 - 326.90}{360} = \frac{33.10}{360} \approx 1/124

    This matches the given relative precision.

To solve these individually:

  • Measured Distance: Each distance is directly given, so you don't need to calculate them.
  • Included Angles: Each angle is between lines at the respective stations. You calculate the sum of the angles to check against the theoretical value.
  • Error Calculation: The difference between the total computed angle and the theoretical angle helps assess the precision.

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Math Problem Analysis

Mathematical Concepts

Surveying
Polygon Interior Angles
Relative Precision

Formulas

Sum of Interior Angles = (n - 2) * 180°
Relative Precision = (Theoretical Sum of Angles - Total Computed Interior Angle) / Theoretical Sum of Angles

Theorems

Polygon Interior Angle Sum Theorem

Suitable Grade Level

College-level or Advanced High School (Grades 11-12)