Runner A is initially 2.4 km west of a flagpole and is running with a constant velocity of 7.6 km/h due east. Runner B is initially 7.6 km east of the flagpole and is running with a constant velocity of 4.4 km/h due west. What will be the distance of the two runners from the flagpole when their paths cross?
they are 10 km apart, and are closing at a rate of 12 kph
they will meet in 5/6 h
pick one and find the distance they travel in 5/6 h , then find the distance from the flagpole
i dont understand
To find the distance of the two runners from the flagpole when their paths cross, we can begin by setting up a coordinate system.
Let's assume the flagpole is at the origin (0,0) on the number line, with east being the positive direction and west being the negative direction.
Runner A starts 2.4 km west of the flagpole, which means their initial position is at (-2.4,0). Runner A is running with a constant velocity of 7.6 km/h due east.
On the other hand, Runner B starts 7.6 km east of the flagpole, which means their initial position is at (7.6,0). Runner B is running with a constant velocity of 4.4 km/h due west.
Now, we need to determine when the two runners will meet or when their paths will cross.
To do that, let's assume t represents the time (in hours) it takes for the paths to cross.
For Runner A, their position (x) at any time t can be expressed as: x_A = -2.4 + 7.6t
For Runner B, their position (x) at any time t can be expressed as: x_B = 7.6 - 4.4t
To find the time when their paths cross, we need to equate the positions of Runner A and Runner B:
-2.4 + 7.6t = 7.6 - 4.4t
Now, we can solve this equation to find the value of t.
-2.4 + 7.6t + 4.4t - 7.6 = 0
Combine like terms:
12t - 10 = 0
Add 10 to both sides:
12t = 10
Divide both sides by 12:
t = 10/12
Simplify:
t = 5/6
So, the paths of Runner A and Runner B will cross after 5/6 hours.
To determine the distance of the two runners from the flagpole when their paths cross, we substitute t = 5/6 into either of the position equations. Let's use Runner A's equation:
x_A = -2.4 + 7.6(5/6)
Simplify:
x_A = -2.4 + 38/6
x_A = -2.4 + 6.33
x_A ≈ 3.93 km
Therefore, when their paths cross, Runner A will be approximately 3.93 km east of the flagpole. The distance of Runner B from the flagpole can be found by substituting t=5/6 into Runner B's position equation, but since Runner B is running in the opposite direction, the distance will be negative. Runner B will be approximately 3.93 km west of the flagpole when their paths cross.