A surveyor measures the distance across a straight river by the following method. Starting directly across from a tree on the opposite bank, she walks d = 72 m along the riverbank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is è = 30.0°. How wide is the river?
30
To find the width of the river, we can use trigonometry. Let's consider the triangle formed by the river, the baseline, and the line of sight to the tree.
We know that the surveyor walked a distance of 72 m along the riverbank, which forms the baseline of the triangle. Let's call this distance "b".
We also know the angle between the baseline and the line of sight to the tree is 30.0°. Let's call this angle "θ".
Now, we can use the trigonometric function tangent (tan) to find the width of the river.
Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the width of the river, and the adjacent side is the baseline. Mathematically, we have:
tan(θ) = opposite / adjacent
tan(θ) = width of the river / baseline
Rearranging the equation, we get:
width of the river = tan(θ) * baseline
Substituting the known values:
width of the river = tan(30.0°) * 72 m
Now, we can calculate the width of the river using a scientific calculator:
width of the river = tan(30.0°) * 72 m
width of the river ≈ 41.569 m
Therefore, the width of the river is approximately 41.569 meters.