The following vertical circle readings were taken by a theodolite set up at station A to observe targets located at P and Q The value of the vertical angle PAQ is _______Instrument at Sighted to Observation - 1 Observation - 2 Vernier C Vernier D Vernier C Vernier D A P $3^\circ 10' 10''$ $10' 20''$ $3^\circ 10' 30''$ $10' 50''$ A Q $-2^\circ 40' 40''$ $41'00''$ $-2^\circ 41' 20''$ $41' 10''$
Calculating Vertical Angle PAQ
This solution calculates the vertical angle PAQ using theodolite readings, averaging measurements from two observations to improve accuracy.
Observed Vertical Circle Readings
| Target | Observation | Vernier C Reading | Vernier D Reading |
|---|---|---|---|
| P | Observation - 1 | $3^\circ 10' 10''$ | $3^\circ 10' 20''$ (Interpreted) |
| Observation - 2 | $3^\circ 10' 30''$ | $3^\circ 10' 50''$ (Interpreted) | |
| Q | Observation - 1 | $-2^\circ 40' 40''$ | $-2^\circ 41' 00''$ (Interpreted) |
| Observation - 2 | $-2^\circ 41' 20''$ | $-2^\circ 41' 10''$ (Interpreted) |
Step 1: Average Readings for Target P
Calculate the mean reading for target P across both observations.
- Mean (Obs 1): $\frac{3^\circ 10' 10'' + 3^\circ 10' 20''}{2} = 3^\circ 10' 15''$
- Mean (Obs 2): $\frac{3^\circ 10' 30'' + 3^\circ 10' 50''}{2} = 3^\circ 10' 40''$
- Overall Mean P: $\frac{3^\circ 10' 15'' + 3^\circ 10' 40''}{2} = 3^\circ 10' 27.5''$
Step 2: Average Readings for Target Q
Calculate the mean reading for target Q across both observations.
- Mean (Obs 1): $\frac{-2^\circ 40' 40'' + (-2^\circ 41' 00'')}{2} = -2^\circ 40' 50''$
- Mean (Obs 2): $\frac{-2^\circ 41' 20'' + (-2^\circ 41' 10'')}{2} = -2^\circ 41' 15''$
- Overall Mean Q: $\frac{-2^\circ 40' 50'' + (-2^\circ 41' 15'')}{2} = -2^\circ 41' 02.5''$
Step 3: Calculate Vertical Angle PAQ
Subtract the overall mean reading for Q from the overall mean reading for P.
Angle PAQ = Overall Mean P - Overall Mean Q
Angle PAQ = $3^\circ 10' 27.5'' - (-2^\circ 41' 02.5'')$
Angle PAQ = $3^\circ 10' 27.5'' + 2^\circ 41' 02.5''$
Angle PAQ = $5^\circ 51' 30''$
Final Result
The vertical angle PAQ is determined to be $5^\circ 51' 30''$.
Important Questions from Theodolite
- Which of the following errors is/are eliminated by taking both left and right face observations in a theodolite?
- When conducting a survey using a total station, a zenith angle is measured as $84°13′56″$ in the direct mode. What is the equivalent zenith angle in the reverse mode?
- "Transit the telescope" of a theodolite involves
- Choose the CORRECT statement(s).
- Face Left and Face Right observations using a vernier theodolite will eliminate
