Final answer:
To determine the change in range, a surveyor typically uses triangulation by measuring angles from two points on a baseline to the target, then applying trigonometry, which often involves using the Pythagorean theorem in conjunction with other trigonometric relations.
Step-by-step explanation:
To determine the difference in range from a known point to a target, a surveyor would use a method similar to the one described in the scenario with the tree across the river. First, establish a baseline and then measure the angles to the tree from two different points along that baseline to create a triangle. From these angles and the known baseline distance, the surveyor can apply the principles of trigonometry. The most effective technique is using triangulation, where these measurements allow the calculation of distances using the sine and cosine rules. If exact coordinates are available, the surveyor could also subtract one set of coordinates from the other (option a), but this isn't a typical field method unless they are working with GPS or other coordinate-based systems. The method of measuring the vertical shift in mils between the known point and the target (option b) is not sufficient on its own because it ignores the horizontal component of the distance. The Pythagorean theorem (option c) can be part of the solution but, it applies to right triangles, whereas not all observation setups form right angles. Lastly, calculating the target's distance from each observation point and subtracting one from the other (option d) can provide the range change if the observations and calculations are precise.