in order to determine the width of a river, a base line of 100 ft is marked off on one side. On the other side of the river is a large rock. When sighted from each end of the base line it is found that the lines of sight make angles of 60 degrees and 75 degrees, respecively, with the base line.Determine the width of the river
Let w be the river's width. Draw a scalene triangle with 100 ft as one side and internal angles of 105 and 60 degrees at each corner formed by that side. The two other sides will correspond to the sight lines. The opposite vertex will be the other side of the river. Use the Law of Sines to get the sides of the triangle. Once you have that, the river width can be obtained from a triangle side length and the sighting angle.
This is not about physics; it is pure trigonometry
To determine the width of the river, we can use trigonometry. Let's break down the problem and follow these steps:
1. Draw a diagram: Start by drawing a diagram of the situation described in the problem. Draw a straight line to represent the river and mark a base line of 100 ft on one side. Place a large rock on the other side of the river.
2. Identify the angles: The problem states that the lines of sight from each end of the base line make angles of 60 degrees and 75 degrees, respectively. Label these angles on the diagram.
3. Determine the triangle: From the diagram, we can see that we have a triangle formed by the base line, the sightline from one end, and the sightline from the other end. Label the sides and angles of the triangle.
4. Use trigonometry: To find the width of the river, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant.
In this case, we can use the following equation:
sin(A) / a = sin(B) / b = sin(C) / c
Where A, B, and C are the angles of the triangle, and a, b, and c are the respective sides.
5. Let's set up the equation: In our triangle, angle A is 60 degrees, angle B is 75 degrees, and side a is 100 ft (the base line). Let side b represent the width of the river that we are trying to find.
We can use the equation as follows:
sin(60) / 100 = sin(75) / b
6. Calculate the width of the river: Plug the values into the equation and solve for b.
sin(60) / 100 = sin(75) / b
0.866 / 100 = 0.966 / b
Cross-multiply and solve for b:
0.866 * b = 100 * 0.966
b = (100 * 0.966) / 0.866
b ≈ 112.36 ft
Therefore, the width of the river is approximately 112.36 ft.