Question

A line, AB, is measured at both ends as follows: Instrument at A: slope distance = 879.209 m; vertical angle = +1°26′50″ Instrument at B: slope distance = 879.230 m; vertical angle = −1°26′38″ The heights of the instrument, reflector, and target are equal for each observation.Compute the horizontal distance AB.#14

Answer 1

Calculate the horizontal distance between two points by converting the vertical angle to decimal degrees and applying the cosine to the slope distance. The equal heights of the instrument, reflector, and target simplify the process, ensuring that either measurement will yield the same horizontal distance.

Step-by-step explanation:

The student has asked how to compute the horizontal distance between two points A and B based on slope distance measurements taken from both ends with their respective vertical angles. This is a trigonometric problem that involves calculating the horizontal component of a slope distance when the vertical angle is known. The heights of the instrument, reflector, and target are equal, which simplifies the problem.

Using the formula for horizontal distance (D) calculated from the slope distance (S) and the vertical angle (α), we have:

1. Convert the vertical angles from degrees, minutes, and seconds to decimal degrees.
2. Calculate the horizontal distance using the cosine of the converted angle and the slope distance (D = S * cos(α)).
3. Since the vertical angles are complementary and the height of instrument and reflector are the same, there won't be any discrepancy due to the different angle measurements from A and B. Meaning, the horizontal distance can be calculated from either point.

To calculate the horizontal distance, it's crucial that we take one of the slope distances and calculate the horizontal component using the corresponding angle.