Runner A is initially 5.3 km west of a flagpole and is running with a constant velocity of 8.1 km/h due east. Runner B is initially 4.7 km east of the flagpole and is running with a constant velocity of 7.8 km/h due west. How far are the runners from the flagpole when their paths crossed? answer in units of km.
i understand how to set this problem up, a little bit, but I cannot figure out what the final answer is
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To find the distance between the runners when their paths crossed, we need to determine the time it took for them to cross paths.
Let's represent the distance travelled by each runner as a function of time.
For Runner A: Distance_A = 5.3 km + (8.1 km/h) * t, where t is the time in hours.
For Runner B: Distance_B = 4.7 km + (7.8 km/h) * t, where t is the time in hours.
To find the time at which their paths crossed, we equate the two distances:
5.3 km + (8.1 km/h) * t = 4.7 km + (7.8 km/h) * t
Simplifying the equation:
5.3 km - 4.7 km = (7.8 km/h - 8.1 km/h) * t
0.6 km = -0.3 km/h * t
Dividing both sides by -0.3 km/h, we get:
t = -0.6 km / -0.3 km/h
t = 2 hours
So, it took 2 hours for the runners to cross paths.
To find the distance of each runner from the flagpole at this time, we substitute t = 2 into the distance equations:
Distance_A = 5.3 km + (8.1 km/h) * 2
Distance_A = 5.3 km + 16.2 km
Distance_A = 21.5 km
Distance_B = 4.7 km + (7.8 km/h) * 2
Distance_B = 4.7 km + 15.6 km
Distance_B = 20.3 km
Therefore, when their paths crossed, Runner A was 21.5 km from the flagpole, and Runner B was 20.3 km from the flagpole.