he 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
he 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.
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.
Next, we need to calculate the difference in height between the EDM and the target. This is given by:
Height of EDM - Height of Reflector = 1.495 m - 1.3 m = 0.195 m
Height of Theodolite - Height of Target = 1.615 m - 1.385 m = 0.23 m
The total difference in height is therefore 0.195 m + 0.23 m = 0.425 m.
The slope distance is given as 1650.452 m.
We can now use the formula for the horizontal distance (d) given the slope distance (S), the vertical angle (A), and the difference in height (h):
d = S * cos(A) - h
Substituting the given values:
d = 1650.452 m * cos(3.042222 degrees) - 0.425 m
This gives the horizontal distance between M and P.
Please note that the cosine function in most calculators requires the angle to be in radians. To convert from degrees to radians, multiply the angle by π/180. So, the angle in radians is 3.042222 degrees * π/180 = 0.053105 radians.
So, the calculation becomes:
d = 1650.452 m * cos(0.053105 radians) - 0.425 m
This will give you the horizontal distance between M and P.
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