To determine how long it will take the two trains to reach each other, we begin by calculating their combined speed. Eitan's train is traveling west at 65 miles per hour, and Dmitri's train is moving east at 55 miles per hour.
Step 1: Calculate the combined speed.
The combined speed of the two trains is: \[ 65 \text{ mph} + 55 \text{ mph} = 120 \text{ mph} \]
Step 2: Determine the initial distance between the two trains.
The initial distance between Eitan and Dmitri is 150 miles.
Step 3: Calculate the time it takes for the two trains to meet.
Using the formula for time, which is distance divided by speed: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{150 \text{ miles}}{120 \text{ mph}} = 1.25 \text{ hours} \]
Step 4: Calculate how far each train travels in that time.
For Eitan's train: \[ \text{Distance for Eitan} = \text{Speed} \times \text{Time} = 65 \text{ mph} \times 1.25 \text{ hours} = 81.25 \text{ miles} \]
For Dmitri's train: \[ \text{Distance for Dmitri} = \text{Speed} \times \text{Time} = 55 \text{ mph} \times 1.25 \text{ hours} = 68.75 \text{ miles} \]
Step 5: Determine how far outside the city they are when they reach each other.
Both Eitan and Dmitri have traveled toward each other, and since they started 150 miles apart, we add their distances traveled: \[ 81.25 \text{ miles (Eitan)} + 68.75 \text{ miles (Dmitri)} = 150 \text{ miles (correct, as they meet)} \]
Since they are traveling towards the city, their meeting point can be interpreted as being either Eitan's distance from the city or Dmitri's distance from the city.
Assuming they start at the city limits:
- Eitan is 81.25 miles west of the city.
- Dmitri is 68.75 miles east of the city.
Finally, the results are:
- The time when the two trains reach each other is 1.25 hours.
- The trains are 81.25 miles away from the city when they reach each other.
Thus, the final answers are:
The time when the two trains reach each other is 1.25 hours.
The trains are 81.25 miles away from the city when they reach each other.
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