Chapter 20, Problem 20.22P

To determine

The combined factor for the line.

The geodetic azimuth.

The ground distance in meters.

The combined factor for the line is $0.999871805$ .

The geodetic azimuth is $5°51^{\prime }57.69^{″}$ .

The ground distance in meters is $565.0527\text{m}$ .

Given:

The SPCS83 state plane coordinates of points A and B are tabulated below.

Point	E(m)	N(m)
A	$170$	$222$
B	$227.7499$	$784.0939$

Calculation:

Write the Equation for Elevation factor.

$\text{Elevation}\text{Factor}=\frac{R}{R+H+N}$ …… (I)

Here, $R$ is average radius of earth, $H$ is the average orthometric height of the area, and $N$ is the geoid height.

Assume, the mean orthometric height as $243\text{m}$ and average geoid height as $-31.273\text{m}$ .

Substitute $6371000\text{m}$ for $R$ , $-31.273\text{m}$ for $N$ , and $243\text{m}$ for $H$ in Equation (I).

$\begin{array}{c}\text{Elevation}\text{Factor}=\frac{6371000\text{m}}{6371000\text{m}+243\text{m}-31.273\text{m}}\\ =\frac{6371000\text{m}}{6371211.727\text{m}}\\ =0.999966768\end{array}$

Calculate the combined factor using formulae.

$\text{Combined}\text{factor}=\text{Elevation}\text{factor}\times \text{Scale}\text{factor}$…… (II)

Substitute $0.9999050344$ for scale factor, and $0.999966768$ for Elevation factor in Equation (II).

$\begin{array}{c}\text{Combined}\text{factor}=0.999966768\times 0.9999050344\\ =0.999871805\end{array}$

Write the expression to calculate the grid distance.

$AB=\sqrt{{\left(E_B-E_A\right)}^2+{(N_B-N_A)}^2}$ …… (III) Substitute $170$ for $E_A$ , $227.7499$ for $E_B$ , $222$ for $N_A$ , and $784.0939$ for $N_B$ in Equation (III).

$\begin{array}{c}AB=\sqrt{{\left(227.7499\text{m}-170\text{m}\right)}^2+{(784.0939\text{m}-222\text{m})}^2}\\ =\sqrt{{\left(57.7499\text{m}\right)}^2+{\left(562.0939\text{m}\right)}^2\text{}}\\ =565.0527\text{m}\end{array}$

Write the Expression to calculate ground distance.

$\text{Grid}\text{distance}=\text{Ground}\text{distance}\times \text{combined}\text{factor}$…… (IV)

Substitute $565.0527\text{m}$ for Grid distance, and $0.999871805$ for combined factor in Equation (IV).

$\begin{array}{l}565.0527\text{m}=\text{Ground}\text{distance}\times 0.999871805\\ \text{Ground}\text{distance}=\frac{565.0527\text{m}}{0.999871805}\\ \text{Ground}\text{distance}=565.7540\text{m}\end{array}$

Calculate the angle of grid line.

${\theta }_{AB}={\tan }^{-1}\left(\frac{N_B-N_A}{E_B-E_A}\right)$…… (V)

Substitute $170$ for $E_A$ , $227.7499$ for $E_B$ , $222$ for $N_A$ , and $784.0939$ for $N_B$ in Equation (V).

$\begin{array}{c}{\theta }_{AB}={\tan }^{-1}\left(\frac{784.0939-222}{227.7499-170}\right)\\ ={\tan }^{-1}\left(\frac{562.0939}{57.7499}\right)\\ ={\tan }^{-1}\left(9.73324\right)\\ =84°8^{\prime }2.31^{″}\end{array}$

The line AB is lies in first quadrant, origin at A. In this case, the azimuth of line can be calculated by subtracting the angle of line from $90°$ , that is,

$\alpha =90-{\theta }_{AB}$

Substitute $84°8^{\prime }2.31^{″}$ for ${\theta }_{AB}$ .

$\begin{array}{c}\alpha =90-84°8^{\prime }2.31^{″}\\ =5°51^{\prime }57.69^{″}\end{array}$

Conclusion:

Thus, the combined factor for the line is $0.999871805$ .

Thus, the geodetic azimuth is $5°51^{\prime }57.69^{″}$ .

Thus, the ground distance in meters is $565.0527\text{m}$ .