Two stations A and B are 2,000m apart. When an airplane was directly above A, an observer at B found that the angle of elevation of the airplane is 60° . How far is the plane from the station B??
as usual, draw a diagram. You can easily see that if the distance is x,
2000/x = cos 60°
To find the distance of the plane from station B, we can use trigonometry and the given angle of elevation.
Let's assume that the plane is at point P, directly above station A. We also need to introduce another point, point Q, which is directly below the plane P and vertically above station B. This forms a right-angled triangle PQB.
We know that the angle of elevation of the plane from station B is 60°. This means that the angle at point Q (angle PBQ) is also 60°, as it is vertically opposite to the angle of elevation. Therefore, we have a right-angled triangle PBQ with an angle of 60°.
Since angle PBQ is a right angle, we can also conclude that angle PQB is 180° - 90° - 60° = 30°.
Now, let's focus on triangle PBQ. We have the side PQ (which is the distance between the plane and station A) and the side BQ (which is the distance between the plane and station B).
We need to find the value of BQ (the distance between the plane and station B). To do this, we can use the trigonometric function tangent (tan):
tan(PQB) = BQ / PQ
Since tangent is the opposite over adjacent:
tan(30°) = BQ / 2000m
We know that tan(30°) = 1 / √3 ≈ 0.5774
0.5774 = BQ / 2000m
Now we can solve for BQ:
BQ = 0.5774 * 2000m
BQ ≈ 1154.8m
Therefore, the plane is approximately 1154.8m away from station B.
To find the distance of the plane from station B, we can use trigonometry and the information given.
Let's denote the distance of the plane from station B as 'x'.
In a right triangle formed by the airplane, station A, and station B, we have the following:
- The side opposite the angle of elevation (60°) is the distance between the airplane and station A, which is 2000m.
- The side adjacent to the angle of elevation is the distance between the airplane and station B, which is 'x'.
- The hypotenuse of the triangle is the straight line distance between station A and station B, which is also 2000m.
Using the trigonometric function tangent (tan), we can relate the angle of elevation (60°) to the side opposite (2000m) and the side adjacent (x):
tan(60°) = opposite/adjacent
tan(60°) = 2000/x
Now we can solve for 'x' by rearranging the equation:
x = 2000 / tan(60°)
Using a calculator, we can find the value of the tangent of 60°, which is approximately 1.732:
x = 2000 / 1.732
Calculating this expression, we find that:
x ≈ 1154.7 meters
Therefore, the plane is approximately 1154.7 meters away from station B.