To solve the problem of calculating the total distance traveled by the bee between two trains, we need to break down the scenario into manageable steps. 

We'll use tables to clearly illustrate the calculations and derive the total distance traveled by the bee.

Problem Breakdown

Train Speeds:

Speed of Train 1: 50 km/h

Speed of Train 2: 70 km/h

Initial Distance Between Trains: 100 km

Speed of the Bee: 80 km/h

Approach

1. Calculate the Time Until Collision:

The two trains are moving towards each other. Therefore, the rate at which the distance between them is decreasing is the sum of their speeds.

2. Calculate the Total Distance Traveled by the Bee:

The bee travels at a constant speed and keeps flying back and forth until the trains collide. To find the total distance, we need to calculate the time it takes for the trains to collide and then use the bee’s speed to find the distance.

Step-by-Step Solution

1. Time Until Collision

The relative speed of the trains is the sum of their speeds:

Relative Speed = Speed of Train 1 + Speed of Train 2

                          = 50km/h + 70km/h

                          =120km/h

The time (𝑡) it takes for the trains to collide is given by:

𝑡 = Initial Distance / Relative Speed

  =  100km/ 120 km/h

  =  5/6 hours

  ≈  0.833 hours

2. Total Distance Traveled by the Bee

The total distance traveled by the bee is the bee's speed multiplied by the time until the trains collide.

Using the time calculated:

Total Distance = Bee’s Speed × 𝑡

                         = 80km/h × 5/6hours

                         = 400/ 6km

                         ≈ 66.67 km

Conclusion

The total distance traveled by the bee, which continuously flies back and forth between the two trains until they collide, is approximately 66.67 kilometers. This result is derived from calculating the time it takes for the trains to meet and using the bee’s constant speed to determine the distance covered in that time.