The height of an EDM set up at M is 1.495 m. The height of a reflector set up at P is 1.30 m. The height of the theodolite at M used to measure the vertical angle is 1.615 m. The height of the target at P on which the vertical sight is taken is 1.385 m. the slope distance after meteorological corrections is 1650.452 m. The measured vertical angle is +3°02'32". What is the horizontal distance between M and P. (
Question
The height of an EDM set up at M is 1.495 m. The height of a reflector set up at P is 1.30 m. The height of the theodolite at M used to measure the vertical angle is 1.615 m. The height of the target at P on which the vertical sight is taken is 1.385 m. the slope distance after meteorological corrections is 1650.452 m. The measured vertical angle is +3°02'32". What is the horizontal distance between M and P. (
Solution
To find the horizontal distance between M and P, we first need to convert the vertical angle from degrees, minutes, and seconds to decimal degrees.
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Convert the vertical angle to decimal degrees: The vertical angle is given as +3°02'32". To convert this to decimal degrees, we use the formula: Degrees + (Minutes/60) + (Seconds/3600) So, 3 + (2/60) + (32/3600) = 3.042222 degrees.
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Calculate the difference in instrument and target heights: The difference in height between the instrument and the target is given by the difference between the height of the EDM setup at M and the height of the target at P. So, 1.615 m (height of theodolite at M) - 1.385 m (height of target at P) = 0.23 m.
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Calculate the height correction for the slope distance: The height correction for the slope distance is given by the difference in height between the instrument and the target, divided by the slope distance, and then multiplied by the cosine of the vertical angle (in radians). So, (0.23 m / 1650.452 m) * cos(3.042222 degrees in radians) = 0.000139 m.
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Apply the height correction to the slope distance: The corrected slope distance is given by the slope distance minus the height correction. So, 1650.452 m - 0.000139 m = 1650.451861 m.
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Calculate the horizontal distance: The horizontal distance is given by the corrected slope distance multiplied by the cosine of the vertical angle (in radians). So, 1650.451861 m * cos(3.042222 degrees in radians) = 1649.781 m.
So, the horizontal distance between M and P is approximately 1649.781 m.
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