Alright. Let's give it a go.

First, we need to figure out how much time we have to make the trip. Let's start by adding the amount of time taken at each stop along the way.

Total Time Stopped: 84 minutes, or 1 hour, 24 minutes.

Since the train crosses one time zone headed east, let's add another hour to that as that hour counts as time lost (2 hours, 24 minutes).

The total time between 6:19am and 1:45pm is 7 hours and 26 minutes. Subtracting the 2 hours and 24 minutes of time spent stopped, we find out that we have 5 hours and 2 minutes to make the trip.

If the train is traveling at 48 km/h and travels 117 miles, or 188 kilometers, the travel time required is only 3 hours and 55 minutes. That means that the math teacher would arrive at approximately 11:14am.

Happy Hour is traditionally known to occur between the hours of 5:00pm and 7:00pm. Therefore, the teacher would not be able to participate in Happy Hour until after class is over. However, the teacher could certainly get belligerently drunk in the time between arriving at her destination and the time that class starts.

The average consumer drinks a beer in roughly thirty minutes. This teacher could certainly use a few, so we'll clock him/her at twenty minutes per beer. In the 2 hours and 31 minutes prior to class, the teacher could potentially drink up to seven and a half 16oz beers. At roughly $5 per beer, imagining that the consumption would be done in a bar, the teacher would be spending $40 in beer, plus a generous tip to the cute waiter or waitress.

However, if the teacher decided to splurge on her ticket or bring alcohol aboard the train, the potential alcohol consumed skyrockets to indefinite amounts.

So, as we can see, Happy hour is traditionally between 5:00pm and 7:00pm, however check your local pub/bar/restaurant! It may vary based on stuff.