Answer:
The value is [tex] k =109.6 \ miles [/tex]
Step-by-step explanation:
The diagram illustrating the question is shown on the first uploaded image
From the question we are told that
The distance from city A to B is AB = 467.3 miles
The bearing from B to C is [tex]\theta_{BC} = N 28.7E[/tex]
The bearing from B to A is [tex]\theta_{BA} = N 60.7E[/tex]
The bearing from A to B is [tex]\theta_{AB} = S60.7W[/tex]
The bearing from A to C is [tex]\theta_{AC} = S79.1W[/tex]
Generally from the diagram
[tex]\theta_A = 180 - 60.7 -79.1[/tex]
=> [tex]\theta_A = 40.2 ^o[/tex]
Also
[tex]\theta_B = 32^o [/tex]
and
[tex]\theta_C = 180 - (\theta_A +\theta_B )[/tex]
=> [tex]\theta_C = 180 - (40.2 + 32 )[/tex]
=> [tex]\theta_C = 107.8 ^o[/tex]
Generally according to Sine Rule
[tex]\frac{BC}{sin (\theta_A)} = \frac{CA}{sin (\theta_B)} =\frac{AB}{sin (\theta_C)}[/tex]
=> [tex]\frac{BC}{sin (40.2)} = \frac{CA}{sin (32)} =\frac{467.3 }{sin (107.8)}[/tex]
So
[tex]\frac{BC}{sin (40.2)} = \frac{467.3 }{sin (107.8)}[/tex]
=> [tex]BC = 316.8 \ miles[/tex]
Also
[tex]\frac{CA}{sin (32)} = \frac{467.3 }{sin (107.8)}[/tex]
[tex]CA = 260 .1 \ miles [/tex]
Generally the additional flyer miles that Adam will receive if he takes the connecting flight rather than the direct flight is mathematically represented as
[tex]k = [CA +BC] - AB [/tex]
=> [tex] k = [260 .1 +316.8]- 467.3 [/tex]
=> [tex] k =109.6 \ miles [/tex]
