A Fighter Jet Pilot's Hair-Raising Encounter
A sunny summer morning, a fighter jet pilot experiences a hair-raising situation. While on a training exercise, she is alerted by the command center that an Unidentified Flying Object (UFO) has entered the airspace and will soon appear on her radar. Some moments later, the jet's radar detects the object a few hundred feet above, moving at other-wordly speeds on a bearing of 60 ∘ 10 ′ (relative to the jet).
In a few milliseconds, the object travels 8.61 miles in a straight line away from the fighter jet. At this point, radars at both the command center and fighter jet begin to mysteriously malfunction. The only data command center is able to collect is that the object has changed bearing and has traveled 5.21 miles (again in a straight line), before disappearing. The jet's radar is able to additionally relay that the last detectable location of the UFO relative to the fighter jet was some distance directly east of the fighter jet.
Everyone is dismayed at the idea that the UFO could be flying undetectable, next to the fighter jet and so they quickly turn to you and your colleague for help, both of who are analysts and trig whizzes.
What mathematical concept can help solve this mystery?
The problem involves the application of trigonometry, specifically the Law of Cosines, to calculate the direct distance from a fighter jet to the last known position of a UFO. By breaking this problem into two movements forming a right triangle, and knowing one angle and the lengths of two sides, we can use the law of cosines to calculate the unknown distance.
How can trigonometry be applied in this scenario?
This question involves the application of trigonometry to solve a real-world situation. The given scenario implies two straight-line movements of an object (UFO) relative to the stationary jet fighter. Initially, the UFO moves 8.61 miles at a tiny angle (60 degrees 10') relative to the east direction. Its second movement is 5.21 miles directly east of the fighter jet.
It seems like a right triangle might be formed here with the fighter jet as the vertex. The UFO's first path as one side (8.61 miles), and the second as the other side (5.21 miles). The angle between them, we can calculate as (90 - 60 ∘ 10 ′), which is 29°50′. We also know that the UFO's last detectable position was due east from the jet, which allows us to use the Law of Cosines to solve for the unknown side of this right triangle, which represents the direct distance (d) from the jet to the last known position of the UFO. The law of cosines can be applied as:
d² = 8.61² + 5.21² - 2 * 8.61 * 5.21 * cos(29°50')
From this equation, we can calculate d to find the desired distance. This represents the application of trigonometric principles in solving real-world problems.