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

The value is
`k =109.6 \ miles`

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

`(BC)/(sin (\theta_A))  = (CA)/(sin (\theta_B)) =(AB)/(sin (\theta_C))`

=>
`(BC)/(sin (40.2))  = (CA)/(sin (32)) =(467.3 )/(sin (107.8))`

So

`(BC)/(sin (40.2))  = (467.3 )/(sin (107.8))`

=>
`BC = 316.8 \ miles`

Also

`(CA)/(sin (32)) =  (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`