Question

The following vertical circle readings were taken by a theodolite set up at station A to observe targets located at P and Q. Find the vertical angle $PAQ$. 

• A. $1^\circ 51' 25''$
• B. $1^\circ 51' 30''$
• C. $5^\circ 51' 25''$
• D. $5^\circ 51' 30''$

Hint:

With a properly adjusted theodolite, take face-left and face-right (two-face) readings and {average} them to remove index error; then difference the means to get the included vertical angle.

Answer 1

The Correct Option is D

Step 1: Mean vertical angle to each target (two-face observation).
Using the Vernier-C readings (face left and face right) to eliminate index error:
For $P$: $v_P=\dfrac{3^\circ10'10''+3^\circ10'30''}{2}=3^\circ10'20''.$
For $Q$: $v_Q=\dfrac{(-2^\circ40'40'')+(-2^\circ41'20'')}{2}=-2^\circ41'00''.$
Step 2: Required angle $PAQ$.
\[ \angle PAQ = v_P - v_Q = 3^\circ10'20'' - (-2^\circ41'00'') = 5^\circ51'20'' \approx 5^\circ51'30''. \] (Nearest choice.)
\[ \boxed{5^\circ 51' 30''} \]