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). Read more

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} ]

2. 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} ]

3. 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.

4. 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} ]

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