A slope distance of 204.834 m was determined by EDM, the instrument and prism having been set up 1.645 m and 1.873 m respectively above their stations. The vertical angle between the stations was later measured as 2° 17' 40", the instrument station being the higher of the two. Calculate the horizontal length of the line.
Question
A slope distance of 204.834 m was determined by EDM, the instrument and prism having been set up 1.645 m and 1.873 m respectively above their stations. The vertical angle between the stations was later measured as 2° 17' 40", the instrument station being the higher of the two. Calculate the horizontal length of the line.
Solution
To calculate the horizontal length of the line, we need to use the formula for the horizontal distance (D) in a slope distance measurement:
D = S * cos θ
where: S = slope distance θ = vertical angle
First, we need to convert the vertical angle from degrees, minutes, and seconds to decimal degrees.
2° 17' 40" = 2 + 17/60 + 40/3600 = 2.29444°
Next, we need to adjust the vertical angle for the height difference between the instrument and the prism. The height difference is 1.873 m - 1.645 m = 0.228 m.
The adjusted vertical angle (θ') is given by:
θ' = θ + atan(h/S)
where: h = height difference
θ' = 2.29444° + atan(0.228/204.834) = 2.29444° + 0.06364° = 2.35808°
Finally, we can calculate the horizontal distance:
D = 204.834 m * cos(2.35808°) = 204.834 m * 0.999396 = 204.634 m
So, the horizontal length of the line is approximately 204.634 m.
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