Cesar Millan and Rue's Running Challenge: How Far Does Rue Run?

To solve this problem, we need to calculate the distance Rue has run for each round trip from the house to Cesar Millan and back.

Let's assume the round trip distance is 'd' km.

When Cesar Millan is 3.0 km from home, Rue starts running towards him with a speed of 3.0 km/h. It will take Rue (3.0 km / 3.0 km/h) = 1 hour to reach Cesar Millan.

In this 1 hour, Cesar Millan is moving towards home at a speed of 2.0 km/h. Therefore, he will cover a distance of (2.0 km/h * 1 hour) = 2.0 km.

At this point, Cesar Millan and Rue are together, and they both start running towards the house. Since they are running together, their combined speed is (3.0 km/h + 2.0 km/h) = 5.0 km/h.

The time taken to cover the remaining distance (3.0 km - 2.0 km) = 1.0 km at a speed of 5.0 km/h is (1.0 km / 5.0 km/h) = 0.2 hours.

Therefore, Rue runs for a total of (1 hour + 0.2 hours) = 1.2 hours per round trip.

The total distance Rue runs can be calculated by multiplying the round trip distance by the number of round trips he makes.

Let's assume Rue makes 'n' round trips.

Total distance Rue runs = n * d

Since Cesar Millan is returning home, he will also reach home after n round trips.

The total time taken by Cesar Millan to reach home is 3.0 km / 2.0 km/h = 1.5 hours.

We can equate the time taken by Rue and Cesar Millan to reach home:

n * 1.2 = 1.5

Simplifying the equation:

n = 1.5 / 1.2

n = 1.25

Since the number of round trips cannot be a fraction, we round down to the nearest whole number.

n = 1

Therefore, Rue makes 1 round trip.

The total distance Rue runs = 1 * d

Since Rue runs back and forth, the round trip distance is twice the distance from Cesar Millan to the house.

d = 2 * 3.0 km = 6.0 km

The total distance Rue runs = 1 * 6.0 km = 6.0 km

Therefore, Rue has run 6.0 kilometers.