Lines of levels are run from Station A to Station B over three different routes. The route length and difference in elevation between Station A and B is given as follows:
Route
Length (km.)
Difference in elevation (m.)
1
6.3
428.635
2
5.7
426.325
3
3.6
429.524
Determine the most probable value of the elevation of Station B if Station A is at elevation 48.254m.
1st: Compute for the weight for each route
\[
W = \frac{1}{L}
\]
\[
W_1 = \frac{1}{6.3}
\]
\[
W_2 = \frac{1}{5.7}
\]
\[
W_3 = \frac{1}{3.6}
\]
2nd: Compute for the summation of W
\[
\sum W = W_1 + W_2 + W_3
\]
\[
\sum W = \frac{10}{63} + \frac{10}{57} + \frac{5}{18}
\]
\[
\sum W = \frac{1465}{2394}
\]
3rd: Multiply W with the difference in elevation
\[
X = W \times DE
\]
\[
X_1 = \frac{1}{6.3} \times 428.635
\]
\[
X_1 = \frac{85727}{1260}
\]
\[
X_2 = \frac{1}{5.7} \times 426.325
\]
\[
X_2 = \frac{17053}{228}
\]
\[
X_3 = \frac{1}{3.6} \times 429.524
\]
\[
X_3 = 119.3122222
\]
4th: Compute for the summation of X
\[
\sum X = X_1 + X_2 + X_3
\]
\[
\sum X = \frac{85727}{1260} + \frac{17053}{228} + 119.3122222
\]
\[
\sum X = 262.1433835
\]
5th:
\[
MPV = \frac{\sum X}{\sum W}
\]
\[
MPV = \frac{262.1433835}{\frac{1465}{2394}}
\]
\[
MPV = 428.3762867
\]
\[
\therefore \text{Elevation at Station B = Station A + MPV}
\]
\[
\text{Most Probable Elevation at Station B = $48.254 + 428.3762867$}
\]
\[
\text{Most Probable Elevation at Station B = $476.6302867$}
\]
\[
\text{Most Probable Elevation at Station B $\approx$ 476.630 $\leftarrow answer$}
\]
