Trignonometry world problem

Question

Two trains leave a station on different tracks that make an angle of 110 degrees with the station as a vertex. The first train travels at an average rate of 50 mph and the second train travels at an average rate of 40 mph. How far apart, to the nearest tenth of a mile, are the trains after 2 hours?

Answer 1

use the law of cosines

draw a triangular diagram with the two adjacent sides being the tracks and the angle at the vertex being 110º

connect the adjacent sides to form a triangle and this distance will be the distance the trains are apart

one side is 50*2=100 miles and the adjacent side is 40*2=80 miles

call the 110º angle "C" and call the side opposite this angle "c"

call the adjacent sides "a" and "b"

c^2=a^2+b^2-2abcosC

c^2=100^2+80^2-2*100*80*cos110º

c^2=10,000+6400-16,000*(-0.3420201)

c^2=16,400+5472.3216

c^2=21,872.322

take the square root of both sides

c=147.89 miles

c=147.9 miles to the nearest tenth