Understanding Reduced Horizontal Distance Calculation

The question asks for the reduced horizontal distance between two points P and Q over the reference ellipsoid. We are given the measured horizontal distance and the average elevation along the line PQ. The reduction accounts for the difference in length between the measured distance (affected by average elevation) and the distance projected onto the ellipsoid.

Calculating the Correction

The correction ($\Delta d$) due to average elevation ($h$) and Earth's radius ($R$) is approximated by the formula:

$ \Delta d \approx \frac{d_{measured} \times h}{R} $

Where:

• $d_{measured}$ = Measured horizontal distance = $12380.56$ m
• $h$ = Average elevation = $748.82$ m
• $R$ = Radius of Earth = $6378$ km = $6378000$ m

Step-by-Step Calculation

1. Convert Radius to Meters: The radius of the Earth $R$ is given as $6378$ km, which needs to be converted to meters: $R = 6378 \, \text{km} \times 1000 \, \text{m/km} = 6378000$ m.
2. Calculate the Correction ($\Delta d$): Substitute the values into the correction formula: $ \Delta d = \frac{12380.56 \, \text{m} \times 748.82 \, \text{m}}{6378000 \, \text{m}} $ $ \Delta d \approx 1.453778 \, \text{m} $
3. Calculate the Reduced Horizontal Distance ($d_{reduced}$): Subtract the correction from the measured distance: $ d_{reduced} = d_{measured} - \Delta d $ $ d_{reduced} = 12380.56 \, \text{m} - 1.453778 \, \text{m} $ $ d_{reduced} \approx 12379.106222 \, \text{m} $
4. Round the Result: Round the reduced horizontal distance to two decimal places as required: $d_{reduced} \approx 12379.11$ m.

The calculated value $12379.11$ m falls within the provided range of 12250 m to 12379.5 m.