Distance: 412.31 km
True Bearing: 284°
Compass Bearing: N14.036°W
We can draw a figure to represent the given plane route. With this information, we can find the plane's distance and bearing from its original starting point. See the attached image.
This creates a right triangle, so we can use the Pythagorean theorem to find the distance, where x is the hypotenuse.
a² + b² = c²
100² + 400² = x²
10,000 + 160,000 = x²
170,000 = x²
[tex]\sqrt{170,000} = x[/tex]
412.31 km ≈ x
Next, we will find the true bearing from its starting point. This will be the angle measured clockwise from north. We will use the trigonometric function tangent to find the needed angle, angle b.
[tex]\displaystyle tan\theta=\frac{\text{oppsite}}{\text{adjacent}} =\frac{100}{400} =\frac{1}{4}[/tex]
[tex]\displaystyle tan^{-1}(tan\theta)=tan^{-1}(\frac{1}{4})[/tex]
[tex]\displaystyle \theta=14.036\°[/tex]
This is the angle at the end point (∠b), but we will use this angle to find the bearing from its starting point. A bearing from the north in a west direction is 270 degrees, so we can add our angle to this to find the correct bearing since our plane is traveling in the northwest direction, moving west first then north.
270° + 14.036° = 284.036° ≈ 284°
We can also write the bearing as compass bearing. In this method, we start with north or south, then turn towards west or east at a certain degree.
Starting at the starting point, we face north and turn 14.036° west to the ending point. This can be written as N14.036°W.
