A plane setts off from airport A and flies due east for 100 km, then turns to fly due south for 80 km before landing at an airport b show that the approximate bearing of airport b from airport a is 129
if the plane turns south at point P, then if you set AP along the +x axis, then the angle θ=PAB is
tanθ = -80/100
θ = -39°
That's E39°S or a bearing of 129°
To determine the approximate bearing of airport B from airport A, we can use the concept of trigonometry and the principles of right-angled triangles.
First, let's visualize the situation described in the question:
```
A
|
|
|
100 km
|
--------------------------------> East
| /
| /
80 km /
| /
| /
| /
\/
South
```
From the diagram, we can see that the distance between airport A and airport B forms the hypotenuse of a right-angled triangle.
Using the Pythagorean theorem, we can calculate the length of the hypotenuse:
c^2 = a^2 + b^2
In this case, a = 100 km (eastward distance) and b = 80 km (southward distance). Let's calculate c:
c^2 = 100^2 + 80^2
c^2 = 10000 + 6400
c^2 = 16400
Taking the square root of both sides, we find:
c ≈ 128.062 km
Now, let's find the angle using trigonometric functions. We want to find the angle opposite to the southward distance (80 km).
We can use the tangent function:
tan(angle) = opposite/adjacent = b/a
tan(angle) = 80/100
tan(angle) = 0.8
Taking the inverse tangent (tan^-1) of both sides to find the angle:
angle = tan^-1(0.8)
angle ≈ 38.66 degrees
To convert the angle to a bearing, we need to add or subtract the angle from the reference direction (east). Since the plane is flying towards the south, we subtract the angle from 90 degrees (east is at 90 degrees).
Bearing = 90 degrees - angle
Bearing ≈ 90 - 38.66
Bearing ≈ 51.34 degrees
To convert the angle to a bearing in terms of directions (north, east, south, west):
Bearing ≈ 180 + 51.34
Bearing ≈ 231.34
Approximately, the bearing of airport B from airport A is 231.34 degrees.