Final answer:
The planes are approximately 642.857 miles from the starting point when the jet plane, traveling at 500 mph, overtakes the propeller plane, which is traveling at 150 mph and had a 3-hour head start.
Step-by-step explanation:
To determine how far from the starting point the planes are, we must first calculate how much distance the propeller plane covers in the 3-hour head start. Since the propeller plane travels at 150 mph, in 3 hours it will cover 450 miles (150 mph * 3 hours).
At the point when the jet plane starts, the propeller plane is already 450 miles ahead. To catch up, the jet plane, traveling at 500 mph, must close this gap. After t hours, the jet will have traveled 500t miles, and the propeller plane will have traveled 150t miles plus the initial 450 miles. The planes meet when these distances are the same, so we set 500t equal to 150t + 450 and solve for t:
500t = 150t + 450
500t - 150t = 450
350t = 450
t = 450 / 350
t = 1.2857 hours (approximately)
We can use the time t to find out how far the jet plane traveled: 500 mph * 1.2857 hours = 642.857 miles (approximately). Therefore, the planes are 642.857 miles from the starting point when the jet plane overtakes the propeller plane.