Two trains leaves the same station at 9:30 am and travel in a straight lines at speed of 54 mph and 75 mph,respectively. If the difference in their directions is 136°, how far apart are they at 6:30 pm?
To find the distance between the two trains at 6:30 pm, we need to determine how far each train has traveled by that time.
First, we need to calculate the time elapsed between 9:30 am and 6:30 pm. This is a difference of 9 hours since there are 12 hours in half a day.
Now, let's calculate the distance each train has traveled during this time:
Train 1:
Speed = 54 mph
Time = 9 hours
Distance = Speed × Time = 54 mph × 9 hours = 486 miles
Train 2:
Speed = 75 mph
Time = 9 hours
Distance = Speed × Time = 75 mph × 9 hours = 675 miles
Now, we have the distances traveled by each train. To find the distance between them, we can use the Law of Cosines because we know the side lengths of the triangle and the angle between them.
Using the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, a and b represent the distances traveled by each train, and c represents the distance between them.
Plugging in the values:
c^2 = 486^2 + 675^2 - 2 * 486 * 675 * cos(136°)
c^2 = 236,196 + 455,625 - 656,100 * cos(136°)
Note: Remember to convert the angle from degrees to radians before using it in the equation.
Simplifying the equation:
c^2 = 691,821 - 656,100 * cos(136°)
Now, calculate the value within the cosine function:
cos(136°) ≈ -0.4439
Substituting this value into the equation:
c^2 = 691,821 - 656,100 * -0.4439
c^2 ≈ 420,382.28
Finally, take the square root of both sides to find the distance between the trains:
c ≈ 648.08 miles
Therefore, the two trains are approximately 648.08 miles apart at 6:30 pm.