Two trains depart simultaneously from the same station. The angle between the two tracks on which they leave is 120 degrees. One train travels at an average speed of 45 mph and the other at 70 mph. How far apart are the trains after three hours?
To solve this problem, we can use the concept of relative velocity.
Let's assume that the trains start at the origin (same station). After three hours, the first train will have traveled a distance of 45 mph * 3 hours = 135 miles, and the second train will have traveled a distance of 70 mph * 3 hours = 210 miles.
Now, we need to find the distance between the two trains after three hours. To do that, we can consider the two trains as two sides of a triangle, and we need to find the length of the third side (distance between the two trains).
Since the angle between the two tracks is 120 degrees, the distance between the trains can be calculated using the law of cosines. The formula for the law of cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C),
where c represents the distance between the trains, a and b represent the distances traveled by each train, and C represents the angle between the two tracks.
In this case, a = 135 miles, b = 210 miles, and C = 120 degrees. Let's calculate the distance (c):
c^2 = (135 miles)^2 + (210 miles)^2 - 2 * (135 miles) * (210 miles) * cos(120 degrees)
c^2 = 22725 + 44100 - 56700 * (-0.5)
c^2 = 22725 + 44100 + 56700 * 0.5
c^2 = 22725 + 44100 + 28350
c^2 = 95175 + 28350
c^2 = 123525
Taking the square root of both sides, we find:
c = √123525
c ≈ 351.09 miles.
Therefore, the two trains are approximately 351.09 miles apart after three hours.