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 calculate the distance each train traveled individually.

Let's start by finding the time difference between 9:30 am and 6:30 pm.

9:30 am to 6:30 pm is a total of 9 hours.

Now, we can calculate the distance each train traveled in 9 hours using the formula:

Distance = Speed × Time

For the first train traveling at a speed of 54 mph:

Distance of Train 1 = 54 mph × 9 hours = 486 miles

For the second train traveling at a speed of 75 mph:

Distance of Train 2 = 75 mph × 9 hours = 675 miles

Now that we have the distances traveled by each train, we can calculate the distance between them.

To do this, we can use the cosine law, which states that in a triangle if you know two sides and the angle between them, you can use the following equation:

c² = a² + b² - 2ab * cos(C),

where c is the distance between the two trains, and a, b are the distances traveled by the first and second train, respectively. C is the angle between the two trains, which is given as 136° in this case.

Plugging in the values, we have:

c² = 486² + 675² - 2 * 486 * 675 * cos(136°)

Now, we can calculate the distance between the two trains:

c ≈ √(486² + 675² - 2 * 486 * 675 * cos(136°))

Using a calculator, we find that c ≈ 825 miles.

Therefore, the two trains are approximately 825 miles apart at 6:30 pm.