Runner A is initially 5.0 km west of a flagpole and is running with a constant velocity of 4.5 km/h due east. Runner B is initially 7.0 km east of the flagpole and is running with a constant velocity of 6.5 km/h due west. How far are the runners from the flagpole when their paths cross?
The distance between them is initially 12 km and decreases at a rate 4.5 + 6.5 = 11 km/h.
They meet after = = 12/11 = 1.091 hours. A will have run 4.909 km east and B will have run 7.091 km east at that time.
They will both be 4.909 - 4.5 = 0.409 km east of the flagpole at that time.
To find the distance between the runners when their paths cross, we need to determine the time it takes for them to meet.
Let's consider the distances traveled by each runner:
For Runner A, we have:
Distance traveled = Velocity * Time
Distance = 4.5 km/h * Time (since Runner A is running east, the velocity is positive)
For Runner B, we have:
Distance traveled = Velocity * Time
Distance = -6.5 km/h * Time (since Runner B is running west, the velocity is negative)
Now, let's set up the equation for the total distance of each runner in terms of time:
Total distance for Runner A = 5.0 km (initial distance) + 4.5 km/h * Time
Total distance for Runner B = 7.0 km (initial distance) + (-6.5 km/h * Time)
For their paths to cross, the total distances must be the same. Therefore, we can set up the following equation:
5.0 km + 4.5 km/h * Time = 7.0 km - 6.5 km/h * Time
To solve for Time, let's rearrange the equation:
4.5 km/h * Time + 6.5 km/h * Time = 7.0 km - 5.0 km
11 km/h * Time = 2.0 km
Time = 2.0 km / 11 km/h
Time ≈ 0.182 hours (rounded to three decimal places)
Now that we have the time it takes for the runners to meet, let's find the distance from the flagpole for each runner using the equation:
Distance = Velocity * Time
For Runner A:
Distance = 4.5 km/h * 0.182 hours
Distance ≈ 0.819 km (rounded to three decimal places)
For Runner B:
Distance = -6.5 km/h * 0.182 hours (the negative sign indicates the direction)
Distance ≈ -1.183 km (rounded to three decimal places)
It is important to note that the negative distance for Runner B means that they are west of the flagpole. To find the absolute distance, we can ignore the negative sign, meaning that the distance from the flagpole for Runner B is approximately 1.183 km.
Therefore, when their paths cross, Runner A is approximately 0.819 km east of the flagpole, and Runner B is approximately 1.183 km west of the flagpole.