The FAA and National Transportation Safety Board (NTSB) have begun to investigate the following incident:

At 4:32 pm on March 3, air traffic controllers in St. Louis lost contact with two approaching planes. American Airlines Flight 264 was 57 nautical miles from St. Louis, approaching from the northeast on a heading of 37^° at a rate of 433 knots. At the same time, United Airlines Flight 117 was 73 nautical miles from St. Louis, approaching from the southwest on a heading of 255^° at a rate of 475 knots. Both planes were flying at an altitude of 32,000 ft.

Contact with the planes was reestablished at 4:41 pm. Though the aircrafts and passengers were never in danger, authorities are concerned about how the pilots and controllers handled the situation.

1. For each aircraft, find a function that gives the distance from St. Louis t hours after 4:32 pm.
2. Find a function that gives the distance between the planes t hours after 4:32 pm. (Hint: Use the Law of Cosines.)
3. Sketch the graph of the function found in Problem 2.
4. At what rate was the distance between the planes decreasing at the instant contact was lost? Should have this suggested that the controllers work quickly to reestablish contact?
5. Determine how close the planes actually got and the time at which they were closest.
6. The FAA requires a minimum separation distance of 5 nautical miles. Would have a slight altitude change for one of the planes helped to prevent a violation of the 5 mile rule?
7. If it was your job to write an official report about this matter, what important information would that report include and what would be your conclusion?