Adam must fly home to city A from a business meeting in city B. One flight option flies directly to city A from B, a distance of about 461.1 miles. A second flight option flies first to city C and then connects to A. The bearing from B to C is N29.8 degrees East, and the bearing from B to A is N60.1 degrees East. The bearing from A to B is S60.1 degrees West, and the bearing from A to C is N78.6 degrees West. How many more frequent flyer miles will Adam receive if he takes the connecting flight rather than the direct flight?
To find the distance Adam would have to travel for each option, we can use the Law of Cosines.
For the direct flight from B to A:
Side a = 461.1 miles
Angle A = 60.1 degrees
Angle B = 90 degrees (since it is a right angle)
Using the Law of Cosines:
b^2 = a^2 + c^2 - 2accosA
b^2 = 461.1^2 + c^2 - 2(461.1)(c)cos(60.1)
b^2 = 212,409 + c^2 - 461.1c(0.5)
b^2 = 212,409 + c^2 - 230.55c
For the connecting flight from B to C to A:
Let x be the distance from B to C and y be the distance from C to A.
Using the Law of Cosines for B to C:
x^2 = 461.1^2 + c^2 - 2(461.1)(c)cos(29.8)
x^2 = 212,409 + c^2 - 461.1c(0.875)
x^2 = 212,409 + c^2 - 402.196c
Using the Law of Cosines for C to A:
y^2 = x^2 + 461.1^2 - 2(x)(461.1)cos(78.6)
y^2 = x^2 + 212,409 - 461.1x(0.44)
y^2 = x^2 + 212,409 - 202.764x
Adding x and y for the total distance for the connecting flight:
Total distance = x + y
Total distance = sqrt(x^2) + sqrt(y^2)
Total distance = sqrt(212,409 + c^2 - 402.196c) + sqrt(x^2 + 212,409 - 202.764x)
The difference in frequent flyer miles between the direct flight and the connecting flight would be the total distance for the connecting flight - 461.1 miles.