Question

A horizontal angle \( \theta \) is measured by four different surveyors multiple times and the values reported are given below. 

\[ \begin{array}{|c|c|c|} \hline \text{Surveyor} & \text{Angle \( \theta \)} & \text{Number of observations} \\ \hline 1 & 36^\circ 30' & 4 \\ 2 & 36^\circ 00' & 3 \\ 3 & 35^\circ 30' & 8 \\ 4 & 36^\circ 30' & 4 \\ \hline \end{array} \] 

The most probable value of the angle \( \theta \) (in degree, round off to two decimal places) is \(\underline{\hspace{1cm}}\). 
 

Hint:

The most probable value of an angle in surveying can be calculated using the weighted average of the reported angles, with weights being the number of observations.

Answer 1

Correct Answer: 36

The most probable value of the angle \( \theta \) is calculated by using a weighted average of the reported angles, weighted by the number of observations: \[ \theta_{\text{probable}} = \frac{\sum (\theta_i \times \text{Number of Observations})}{\sum (\text{Number of Observations})}. \] Substituting the values: \[ \theta_{\text{probable}} = \frac{(36^\circ 30' \times 4) + (36^\circ 00' \times 3) + (35^\circ 30' \times 8) + (36^\circ 30' \times 4)}{4 + 3 + 8 + 4}. \] The final value is approximately \( \boxed{36} \) degrees.