Final answer:
To approximate the distance across a river, trigonometry is used by taking the tangent of the angle from a baseline, then solving for the width of the river, which is then converted to feet. The result is approximately 229.67 feet, which does not match any of the options in the provided question.
Step-by-step explanation:
To find the approximate distance across a field when given an angle from a baseline, we need to use trigonometry. If the surveyor walks 100 m along the river and sights an angle of 35° to a tree across the river, we can set up a trigonometric ratio using the tangent function, because we have the angle and the adjacent side of the right triangle formed by the baseline, the line of sight, and the distance across the river.
Using the formula tan(\theta) = \frac{opposite}{adjacent}, where \(\theta\) is the angle of interest, we get:
tan(35°) = \frac{opposite}{100 m}
We can then solve for the opposite side, which is the width of the river:
opposite = 100 m \times tan(35°)
Calculating this we find:
opposite = 100 m \times 0.7002 \approx 70.02 m
To convert meters to feet, we use the conversion factor 1 m = 3.281 ft:
70.02 m \times 3.281 ft/m \approx 229.67 ft
Therefore, the approximate distance across the river is approximately 229.67 feet, which is not one of the options provided in the original question.