Trig and word problems

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 456.2 miles. A second flight option flies first to city C and then connects to A. The bearing from B to C is N29.8°​E, and the bearing from B to A is

N62.8°E. The bearing from A to B is S62.8°​W, and the bearing from A to C is N78.7°W.

How many more frequent flyer miles will Adam receive if he takes the connecting flight rather than the direct​ flight?

Answer 1

First you need to establish were your points A, B, and C go. In order to do that you need to know the direction. It states that the bearing B to C is N29.8^oE which means starting from the origin make a line towards the East (right direction) at an angle 29.8 (the angle should start from North which represents zero degrees), so the line should be in the 1st quadrant. The starting point (at the origin) is point B and where you end the line it is where point C is located, but still don't know exactly the length yet. Again the angle is measured from North going towards East. Now again they give you the bearing of B to A so again make a line starting from B at an angle 62.8 towards the East (the angle is measured between North towards East), so the line should be in the 1st quadrant as well (the end of that line is where point A is located). Now all that is left is to connect points A and C but should be at an angle 78.7 starting from point A going towards the West. Now, we have a triangle ABC and we have certain angles but not the angles A, B, and C yet. Just doing a little adding and subtracting we get

Angles should be

A=38.5^o , B= 33^o , C= 108.5^o

and these angles should all add up to 180^o because it is a triangle.

The bearing from A to B S62.8^oW is just more information to get your angle for A.

We know that line AB = 456.2 miles so we NOW know 1 side and 3 angles so we can apply the Law of Sine

LAW OF SINE:

A/sin(A)=B/sin(B)=C/sin(C)

so it's a simple plug into the equation and should end up with line BC=299.5 miles and AC=262.0 miles

So if you were to take the direct flight from B to A it is 456.2 miles and if you took the flight where you make an extra stop at the city (point C) the total flight would be 561.5 miles.

561.5 mi - 456.2 mi= 105.3 miles

So you will save yourself 105.3 miles if you take the direct flight from the business meeting (point B) and going back home (point A).