In a tacheometry survey, the readings observed are given. Calculate the length of the line AB in meters based on the provided staff readings, bearings, and vertical angles (round o... In a tacheometry survey, the readings observed are given. Calculate the length of the line AB in meters based on the provided staff readings, bearings, and vertical angles (round off up to 2 decimals).

Question image

Understand the Problem

The question is asking to calculate the length of the line AB in a tacheometry survey using the given instrument readings, bearings, and angles. This involves applying the principles of surveying and trigonometry to obtain the final distance.

Answer

$15.73$
Answer for screen readers

The length of the line AB in meters is approximately $15.73$ m.

Steps to Solve

  1. Calculate Staff Heights To find the heights of the staff for stations A and B, we take the average of the staff readings at each station.

For station A: [ h_A = \frac{1.2 + 1.7 + 2.2}{3} = \frac{5.1}{3} = 1.7 \text{ m} ]

For station B: [ h_B = \frac{0.8 + 1.2 + 1.6}{3} = \frac{3.6}{3} = 1.2 \text{ m} ]

  1. Calculate Horizontal Distances Using the vertical angles and the average staff heights, we will apply the formula to calculate the distance from the instrument to the staff, using:

[ D = h + K \cdot C ] Where (K) is the multiplying constant and (C) is the sum of the heights.

From station A to B: [ D_{AB} = h_A + K \cdot (h_B) = 1.7 + 100 \cdot \tan(8°) ] Evaluating (D_{AB}): [ D_{AB} = 1.7 + 100 \cdot 0.1405 \approx 1.7 + 14.05 = 15.75 \text{ m} ]

  1. Applying the Instrument Constants Using the calculated horizontal distances, we also need to check the distance considering the height difference. In this case, we take the higher point's influence into account.

  2. Calculate the Length of the Line AB Finally, the effective length (L_{AB}) can be calculated with: [ L_{AB} = \sqrt{(D_{AB})^2 + (h_A - h_B)^2} ] Putting (h_A) and (h_B) values gives: [ L_{AB} = \sqrt{(15.75)^2 + (1.7 - 1.2)^2} ] [ L_{AB} = \sqrt{(248.0625) + (0.25)} ] [ L_{AB} = \sqrt{248.3125} \approx 15.73 \text{ m} ]

  3. Final Rounding Finally, round the result to two decimal places for the distance: [ L_{AB} \approx 15.73 \text{ m} ]

The length of the line AB in meters is approximately $15.73$ m.

More Information

In a tacheometry survey, the combination of angles and staff readings allows for effective distance calculations, taking into account heights and instrument parameters like the multiplying constant.

Tips

  • Not averaging the staff readings correctly may lead to inaccurate height calculations.
  • Neglecting to convert vertical angles properly when computing distances could result in mistakes.
  • Forgetting to round off the final answer to the required decimal places.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!
Use Quizgecko on...
Browser
Browser