Runner A is initially 6.0 km west of a flagpole and is running with a constant velocity of 7.8 km/h due east. Runner B is initially 4.2 km east of the flagpole and is running with a constant velocity of 7.6 km/h due west. What will be the distance of the two runners from the flagpole when their paths cross?

After t hours, the total distance is covered, so

7.8t+7.6t = 6.0+4.2
t = 0.662 hr

So, A's final position after that long is

-6 + 7.8 * 0.662 = -0.836 km

or, 836 meters west of the pole.

To find the distance of the two runners from the flagpole when their paths cross, we need to determine the time it takes for their paths to intersect.

Let's denote runner A's position with respect to time as xA(t) and runner B's position as xB(t).

Runner A's position can be represented as:
xA(t) = -6.0 km + (7.8 km/h) * t

Runner B's position can be represented as:
xB(t) = 4.2 km - (7.6 km/h) * t

To find the time when their paths cross, we need to solve the equation xA(t) = xB(t):
-6.0 km + (7.8 km/h) * t = 4.2 km - (7.6 km/h) * t

Simplifying the equation:
15.4 km/h * t = 10.2 km
t = 10.2 km / 15.4 km/h
t ≈ 0.6623 hours

Now, we can substitute this value of t back into one of the equations to find the distance from the flagpole when their paths cross.

Using xA(t):
xA(t) = -6.0 km + (7.8 km/h) * 0.6623 hours
xA(t) ≈ -6.0 km + 5.16294 km
xA(t) ≈ -0.83706 km

Therefore, when their paths cross, runner A will be approximately 0.83706 km east of the flagpole.

Using xB(t):
xB(t) = 4.2 km - (7.6 km/h) * 0.6623 hours
xB(t) ≈ 4.2 km - 5.03438 km
xB(t) ≈ -0.83438 km

Therefore, when their paths cross, runner B will be approximately 0.83438 km west of the flagpole.

To find the distance of the runners from the flagpole when their paths cross, we add the distances of both runners from the flagpole:
Distance = |xA(t)| + |xB(t)|
Distance ≈ 0.83706 km + 0.83438 km
Distance ≈ 1.67144 km

Therefore, when their paths cross, the distance of the two runners from the flagpole will be approximately 1.67144 km.