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## The bearing from city A to city B

The bearing from city A to city B is S 40 degrees E and the bearing from city B to city C is N 25 degrees E. It takes 2.1 hours for a car traveling at 54 miles per hour to go from A to B and 1 hours to go from B to C.
Find the distance between city A and city C.

In this problem, you are supposed to use the Law of Cosines, a generalization of the Pythagorean theorem. Pythagorean gives you the third side of a right triangle; the Law of Cosines does the same for any triangle, if you know the angles. It states
c2 = a2 + b2 -2ab cos θ,
where θ is the angle between sides a and b. Notice that if θ=90°, cos θ=0, and you get back the Pythagorean theorem.
In this problem, the three sides of the triangle are the distances between A and B (=a), B and C (=b), and C and A (=c, unknown). We have a= 54*2.1=113.4 miles and b = 54*1=54 miles. The angle between cities A and C, as measured at B, is θ=180-40-25=115°. Therefore,

c2= 113.42+542-2(113.4)(54)cos(115) = 20951,

c = 145 miles.

Hi Dalia;

distance=[(rate)(time)]+[(rate)(time)]
distance=[(54 miles/hour)(2.1 hours)]+[(54 miles/hour)(1 hour)]
Let's cancel units...
Hours is in both the numerators and denominators of both bracketed equations...
distance=[(54 miles/hour)(2.1 hours)]+[(54 miles/hour)(1 hour)]
distance=[(54 miles)(2.1)]+[(54 miles)(1)]
The only units remaining are miles, which is what we want...
distance=(113.4+54) miles
distance=167.4 miles

I do not know why the bearings are provided.