Trigonometric levelling was carried out from two stations P and Q to find the reduced level (R. L.) of the top of hillock, as shown in the table. The distance between Stations P and Q is 55 m. Assume Stations P and Q, and the hillock are in the same vertical plane. The R. L. of the top of the hillock (in m) is _______________________ (round off to three decimal places).
Station
Vertical angle of the top of hillock
Staff reading on benchmark
R. L. of benchmark
P
18∘45′
2.340 m
100.000 m
Q
12∘45′
1.660 m
Correct answer
137.5 to 137.73
Solution
1.Calculate Height of Instrument (HI) at each station:
HIP=R. L. of benchmark+Staff reading at P=100.000+2.340=102.340 m
HIQ=R. L. of benchmark+Staff reading at Q=100.000+1.660=101.660 m
2. Define variables:
Let D be the horizontal distance from station P to the hillock.
The distance from station Q to the hillock is D+55 m (assuming Q is further away based on the smaller angle).
Let hP be the height of the hillock top above the line of sight at P: hP=Dtan(18∘45′)=Dtan(18.75∘)
Let hQ be the height of the hillock top above the line of sight at Q: hQ=(D+55)tan(12∘45′)=(D+55)tan(12.75∘)
3. Equate the R. L. of the top of the hillock:
R.L.top=HIP+hP=HIQ+hQ
102.340+Dtan(18.75∘)=101.660+(D+55)tan(12.75∘)
102.340+0.33945D=101.660+(D+55)×0.22628
102.340+0.33945D=101.660+0.22628D+12.4454
(0.33945−0.22628)D=101.660+12.4454−102.340
0.11317D=11.7654
D=0.1131711.7654≈103.962 m
4. Calculate final R. L. of the top:
R.L.top=102.340+103.962×tan(18.75∘)
R.L.top=102.340+35.289=137.629 m
Rounding to three decimal places, the R. L. is 137.629 m, which falls within the official range of 137.500 to 137.730.
