Question

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 467.3 miles. A second flight option flies first to city C and then connects to A. The bearing from B to C is N28.7​E, and the bearing from B to A is N60.7E. The bearing from A to B is S60.7​W, and the bearing from A to C is N79.1W. How many more frequent flyer miles will Adam receive if he takes the connecting flight rather than the direct​ flight?

Answer 1

Answer:

The value  is  $k =109.6 \ miles$

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  $\theta_{BC} = N 28.7E$

   The bearing from B to  A is  $\theta_{BA} = N 60.7E$

   The  bearing from A to  B is   $\theta_{AB} = S60.7W$

    The  bearing from A to  C is   $\theta_{AC} = S79.1W$

Generally from the diagram

     $\theta_A = 180 - 60.7 -79.1$

=>  $\theta_A = 40.2 ^o$

Also

     $\theta_B = 32^o$

and  

      $\theta_C = 180 - (\theta_A +\theta_B )$

=>   $\theta_C = 180 - (40.2 + 32 )$

=>   $\theta_C = 107.8 ^o$

Generally according to Sine Rule

     $\frac{BC}{sin (\theta_A)} = \frac{CA}{sin (\theta_B)} =\frac{AB}{sin (\theta_C)}$

=>   $\frac{BC}{sin (40.2)} = \frac{CA}{sin (32)} =\frac{467.3 }{sin (107.8)}$    

So

     $\frac{BC}{sin (40.2)} = \frac{467.3 }{sin (107.8)}$

=>  $BC = 316.8 \ miles$

Also  

    $\frac{CA}{sin (32)} = \frac{467.3 }{sin (107.8)}$

    $CA = 260 .1 \ miles$

Generally the additional flyer miles that Adam will receive if he takes the connecting flight rather than the direct​ flight is mathematically represented as

      $k = [CA +BC] - AB$

=>     $k = [260 .1 +316.8]- 467.3$

=>  $k =109.6 \ miles$