Runner A is initially 7.0 km west of a flagpole and is running with a constant velocity of 6.5 km/h due east. Runner B is initially 8.0 km east of the flagpole and is running with a constant velocity of 7.5 km/h due west. How far are the runners from the flagpole when their paths cross?
The distance between the runners decreases at a rate of 14.0 km/h. Since they are initially 15 km apart, they meet after 15/14 = 1.0714 hours
Runner B will have run 1.0714*7.5 = 8.0357 miles at that time. He ends up 0.0357 miles (189 feet) west of the flagpole. So does Runner A.
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To find the distance between the runners when their paths cross, we need to determine the time it takes for them to meet and then calculate the distance each runner has traveled during that time.
First, let's determine the time it takes for the runners to meet. To do this, we need to find the time it takes for Runner A to travel a distance that equals the initial separation between them plus the distance Runner B has traveled.
The initial separation is 7.0 km + 8.0 km = 15.0 km. Since Runner A is running east and Runner B is running west, their distances are added.
Runner A's distance = Runner A's velocity × time
Runner B's distance = Runner B's velocity × time
Since they meet when their distances sum up to 15.0 km, we can write an equation:
Runner A's velocity × time + Runner B's velocity × time = 15.0 km
Substituting the given values, we get:
6.5 km/h × time + (−7.5 km/h) × time = 15.0 km
Now we can solve for time:
−1.0 km/h × time = 15.0 km
time = 15.0 km / (−1.0 km/h)
time = −15.0 h
The negative sign indicates that the runners are moving towards each other.
Since time cannot be negative, let's take the absolute value (magnitude) and keep it positive:
time = 15.0 h
Now, let's find the distance each runner has traveled during this time.
Distance traveled by Runner A = Runner A's velocity × time
Distance traveled by Runner A = 6.5 km/h × 15.0 h
Distance traveled by Runner A = 97.5 km
Distance traveled by Runner B = Runner B's velocity × time
Distance traveled by Runner B = (−7.5 km/h) × 15.0 h
Distance traveled by Runner B = −112.5 km (negative sign indicates opposing direction)
Since distance cannot be negative, let's take the magnitude and keep it positive:
Distance traveled by Runner B = 112.5 km
Finally, to find the distance between the runners when their paths cross, we sum up the two distances:
Distance between the runners = Distance traveled by Runner A + Distance traveled by Runner B
Distance between the runners = 97.5 km + 112.5 km
Distance between the runners = 210.0 km
Therefore, the runners are approximately 210.0 km from the flagpole when their paths cross.