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Two airliners, A and B, fly from point P to points Q and R respectively. Given their coordinates and speed, determine which airliner arrives earlier and by how much.
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4 stepsHere's the solution to question 30.
The problem asks us to determine which airliner arrives at its destination earlier and by how long, given their coordinates and speeds.
Given:
Towns: , , .
Airliner A: Flies from to using the shortest route possible. Speed is .
Airliner B: Takes off from (which is ) at the same time as A and flies to town along the parallel of latitude. Speed is .
We need to calculate the time taken by each airliner. The radius of the Earth is approximately .
Calculate the distance for Airliner A (shortest route).
The shortest route between two points on Earth is along a great circle. The angle between two points and on a great circle is given by:
For Airliner A, and .
, .
(West), (East).
The difference in longitude is .So, .
Convert to radians: radians.
The distance .Time taken by Airliner A: .
Calculate the distance for Airliner B (along the parallel of latitude).
Airliner B flies from to along the parallel of latitude .
The distance along a parallel of latitude is given by , where is the difference in longitude in radians.
Here, .
, .
The difference in longitude .
Convert to radians: radians.Time taken by Airliner B: .
Compare arrival times.
Airliner B arrives earlier.Calculate the difference in arrival time in minutes.
Difference in time .
Convert to minutes: .
Rounding to the nearest minute, the difference is .Airliner B arrives earlier by approximately 167 minutes.
The final answer is .
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