Final answer:
Using the Pythagorean theorem, calculating the distance each train has traveled by 3 p.m. reveals that Train A has traveled 150 miles and Train B has traveled 120 miles. The hypotenuse of the triangle formed, representing the distance between the trains, is approximately 192 miles.
Step-by-step explanation:
To find out how far apart the trains are at 3 p.m., we need to determine the distance each train has traveled since their departure and then use the Pythagorean theorem to find the distance between them. Train A, traveling due north at 50 miles per hour (mi/h), would have been traveling for 3 hours by 3 p.m., which means it has covered 150 miles (because 50 mi/h * 3 h = 150 miles). Train B, traveling due east at 60 mi/h, would have been traveling for 2 hours by 3 p.m. (since it left an hour later), covering 120 miles (because 60 mi/h * 2 h = 120 miles).
Since the trains are moving at right angles to each other, we can imagine a right triangle where Train A has traveled the vertical leg and Train B has traveled the horizontal leg. Thus, to find the hypotenuse (the distance between the trains), we use the Pythagorean theorem (a2 + b2 = c2), where 'a' is 150 miles, 'b' is 120 miles, and 'c' is the distance between the two trains:
- a2 = 1502 = 22500
- b2 = 1202 = 14400
- c2 = a2 + b2 = 22500 + 14400 = 36900
- c = √36900 ≈ 192.19 miles
Therefore, at 3 p.m., the trains are approximately 192 miles apart, which is not an option in the multiple choices provided in the question, suggesting a possible typo or error in the options listed.