Runner A is initially 5.3km west of a flagpole and is running with a constant velocity of 8.8km/hr due east. Runner B is initially 4.2km east of the flagpole and is running with a constant velocity of 7.9km/hr due west. How far are the runners from the flagpole when their paths cross in units of km?
d1 + d2 = 5.3+4.2 = 9.5 km = Distance
between the runners.
8.8t + 7.9t = 9.5
16.7t = 9.5
t = 0.569 h. to cross paths.
d1 = 5.3 - 8.8*0.569 = 0.293 km West of flagpole.
d2 = 4.2 - 7.9*0.569 = -0.295 km West of
Flagpole.
So they meet at approximately 0.294 km
west of flagpole.
To find the distance between the runners and the flagpole when their paths cross, we need to determine the time it takes for their paths to intersect. We can find this time by setting up an equation based on the relative velocities of the runners.
Let's call the time it takes for the paths to intersect t.
For Runner A, distance = velocity x time
Therefore, the position of Runner A at time t is given by:
Position_A = 5.3 km + (8.8 km/hr) x t (since Runner A is running east)
For Runner B, distance = velocity x time
Therefore, the position of Runner B at time t is given by:
Position_B = 4.2 km + (-7.9 km/hr) x t (since Runner B is running west, the velocity is negative)
To find the time t when the positions of the runners are equal, we need to set Position_A equal to Position_B and solve for t:
5.3 km + (8.8 km/hr) x t = 4.2 km + (-7.9 km/hr) x t
Let's simplify the equation:
5.3 km + 8.8 km/hr x t = 4.2 km - 7.9 km/hr x t
Rearranging the terms to isolate t:
8.8 km/hr x t + 7.9 km/hr x t = 4.2 km - 5.3 km
Combining like terms:
16.7 km/hr x t = -1.1 km
Dividing both sides by 16.7 km/hr:
t = -1.1 km / 16.7 km/hr
Simplifying:
t ≈ -0.0659 hr
Since we are dealing with time, we can ignore the negative value and take the absolute value of t as time cannot be negative.
t ≈ 0.0659 hr
Now, let's substitute this value of t back into either Position_A or Position_B to find the distance from the flagpole when their paths cross.
Using Position_A:
Position_A = 5.3 km + (8.8 km/hr) x (0.0659 hr)
Position_A ≈ 5.3 km + 0.5789 km
Position_A ≈ 5.879 km
Therefore, when their paths cross, both runners are approximately 5.879 km from the flagpole.