Question

In a plane triangle, the observed angles \( P, Q \) and \( R \), assumed uncorrelated, with given weights are:

\[ \begin{aligned} P &= 40^\circ 19' 02'' \quad \text{weight} = 1 \\ Q &= 70^\circ 30' 01'' \quad \text{weight} = 2 \\ R &= 69^\circ 11' 00'' \quad \text{weight} = 1 \end{aligned} \]
The most probable values of these angles \( (\hat{P}, \hat{Q}, \hat{R}) \) will be given by:

• A. \( \hat{P} = 40^\circ 19'00'',\ \hat{Q} = 70^\circ 30'00'',\ \hat{R} = 69^\circ 11'00'' \)
• B. \( \hat{P} = 40^\circ 19'01'',\ \hat{Q} = 70^\circ 30'00'',\ \hat{R} = 69^\circ 10'59'' \)
• C. \( \hat{P} = 40^\circ 19'0.8'',\ \hat{Q} = 70^\circ 30'0.4'',\ \hat{R} = 69^\circ 10'58.8'' \)
• D. \( \hat{P} = 40^\circ 18'58.4'',\ \hat{Q} = 70^\circ 30'0.4'',\ \hat{R} = 69^\circ 11'1.2'' \)

Hint:

In angle adjustment problems using least squares, distribute the correction inversely proportional to the weights. Use Lagrange multipliers for optimal adjustment while satisfying the angle sum constraint.

Answer 1

The Correct Option is C

In a plane triangle, the sum of angles must be \( 180^\circ \). The observed sum is:

\[ \begin{aligned} P + Q + R &= (40^\circ 19'02'') + (70^\circ 30'01'') + (69^\circ 11'00'') \\ &= 180^\circ 00'03'' \end{aligned} \]
So, there's an excess of \( +3'' \). We apply least squares adjustment considering the weights to distribute the correction \( -3'' \) among the angles.

Let corrections be \( -x,\ -y,\ -z \) for \( P, Q, R \) respectively. The condition is:

\[ x + y + z = 3'' \]
Using weights \( w_P = 1,\ w_Q = 2,\ w_R = 1 \), we minimize:

\[ \phi = w_P x^2 + w_Q y^2 + w_R z^2 = x^2 + 2y^2 + z^2 \]
Using the method of Lagrange multipliers, the minimum occurs when:

\[ x = z = \lambda, \quad y = \frac{\lambda}{2} \]
So,

\[ x + y + z = \lambda + \frac{\lambda}{2} + \lambda = \frac{5\lambda}{2} = 3'' \Rightarrow \lambda = \frac{6}{5} = 1.2'' \]
\[ x = z = 1.2'',\quad y = 0.6'' \]
Thus, the adjusted angles are:

\[ \hat{P} = 40^\circ 19'02'' - 1.2'' = 40^\circ 19'0.8'' \]
\[ \hat{Q} = 70^\circ 30'01'' - 0.6'' = 70^\circ 30'0.4'' \]
\[ \hat{R} = 69^\circ 11'00'' - 1.2'' = 69^\circ 10'58.8'' \]