Two observers who are 5 km apart simultaneously sight a small airplane flying between them. One observer measures a 51.0-degree inclination angle, while the other observer measures a 40.5-degree inclination angle. At what altitude is the
airplane flying?
To find the altitude at which the airplane is flying, we can use the concept of trigonometry and the information given.
Let's assume that the inclined angles measured by the observers are with respect to the horizontal line. We can create a diagram for visualization:
```
A
/|
c / | b
/ |
B /_____|
a C
```
In the diagram:
- Point A represents the airplane's position.
- Points B and C represent the two observers' positions.
- Lines a, b, and c represent the distances between the points, with c being the distance between the observers.
We are given that the distance between the two observers (c) is 5 km. The observer at point B measures an inclination angle of 51.0 degrees, and the observer at point C measures an inclination angle of 40.5 degrees.
To find the altitude, we need to determine the length of line a (altitude) in the diagram.
Now, let's use trigonometry to find line a:
- In triangle ABC, we can use the tangent function to relate the angle measurements and distances:
- tan(51.0) = a / c (for observer at B)
- tan(40.5) = a / c (for observer at C)
As both equations are equal, we can set them equal to each other:
- tan(51.0) = tan(40.5) = a / c
Now, we can rearrange the equation and solve for a:
- a = c * tan(51.0) = 5 km * tan(51.0)
Using a calculator, we find that tan(51.0) ≈ 1.340191745 corresponds to approximately 5 km * 1.340191745 ≈ 6.700958725 km.
Therefore, the airplane is flying at an altitude of approximately 6.700958725 km.