A train leaves the station at 9:00 pm traveling east at 36 miles per hour. A second train leaves the station at 10:00 pm traveling west at 42 miles per hour. At what time will both trains have traveled the same distance? I need help solving it :)
To solve this problem, we need to find the time at which both trains have traveled the same distance.
Let's start by determining the distance the first train travels before the second train leaves the station. The first train has a head start of 1 hour (since it leaves at 9:00 pm and the second train leaves at 10:00 pm). At a speed of 36 miles per hour, the first train would have traveled 36 miles during this hour.
Now, let's consider the time at which both trains have traveled the same distance. Since the second train is traveling in the opposite direction, it will catch up to the first train at a combined speed of 36 + 42 = 78 miles per hour (the sum of their individual speeds).
To determine the time it takes for the second train to catch up, we divide the distance traveled by the combined speed. Since the second train catches up to the first train, they will have traveled the same distance, so we can use the distance traveled by the first train (36 miles) as our reference point.
Time = Distance / Speed
Time = 36 miles / 78 miles per hour
Time ≈ 0.46 hours (rounded to two decimal places)
Since we know the second train leaves the station at 10:00 pm, we can convert the decimal into minutes:
0.46 hours * 60 minutes = 27.6 minutes
Therefore, both trains will have traveled the same distance at approximately 10:27 pm.
To solve this problem, we can use the concept of relative velocity.
Let's say the time when both trains have traveled the same distance is t hours after the first train leaves the station.
- The first train starts at 9:00 pm and travels east at a speed of 36 miles per hour. So, it covers a distance of 36t miles.
- The second train starts at 10:00 pm, one hour after the first train, and travels west at a speed of 42 miles per hour. So, it covers a distance of 42(t - 1) miles.
To find the time when both trains have traveled the same distance, we need to set the two distances equal to each other:
36t = 42(t - 1)
Now let's solve the equation step-by-step:
1. Distribute the 42 on the right side:
36t = 42t - 42
2. Subtract 36t from both sides to isolate the variable:
36t - 36t = 42t - 42 - 36t
0 = 6t - 42
3. Add 42 to both sides:
42 = 6t
4. Divide both sides by 6 to solve for t:
42/6 = 6t/6
7 = t
Therefore, both trains will have traveled the same distance after 7 hours, which means they will meet at 9:00 pm + 7 hours = 4:00 am.
So, both trains will have traveled the same distance at 4:00 am.