September 5, 2002

Calculus I Student
Franklin & Marshall College
Lancaster, PA 17604

Dear Calculus Student,

I am writing to ask for your assistance in contesting a speeding ticket. I would have written to your professor, Dr. Crannell, but I did not wish to disturb anybody so important. Since you are just a student, I am certain that you will have free time to devote yourself to my request.

Perhaps I should introduce myself. I am G. Olson Overby-Fitzpatrick, the owner and manager of F.A.R. (Fully Automated Rail Trains). We run a train service between the towns of Shoe Chew and Sugar. When I first bought this business, the operation was in terrible shape. (Back then it was called the "Shoe Chew-Sugar Train", which begged the nickname "choo-choo-choo train". I think that "F.A.R. Trains" is much more dignified). The trains were old and in need of serious repair, and after consulting with several expert mechanics, we decided to replace the trains rather than repair them. The rails themselves are also slightly worn, and for that reason state regulations require that we keep our trains' speeds at or below 70 m.p.h.--and this is the supposed cause of our ticket.

The two new engines that we bought are both Fully Automatic--hence our company's name. The engineer driving the train need only push the "go" button, and the doors close, the train pulls out of the station, it maintains a steady speed during the whole 60 mile trip to Sugar, it pulls into the station in Sugar, stops gently, and the doors open. The system is faultless.

No doubt you are wondering how, with a faultless system like ours, it is possible that my engineer Pete got a speeding ticket. Politics, plain and simple.

The fellow I bought the company off of has a cousin, Officer Kovalevskia, who happens to be a police officer. She claims to have clocked the F.A.R. train going 75 m.p.h., which is above the posted speed limit of 70 m.p.h. I think that she's just trying to get back at us for supposedly taking advantage of her brother. And I think I have the evidence to prove it.

Here's what happened. On the day in question, my engineer Pete was testing out the system. (There were no passengers on board, rest assured). He hit two buttons at once: the "go" button and the "slow" button. We wanted to see what would happen. There's another button called "decel", which we know can slow the train from 60 to 0 m.p.h. in 12 seconds, but we weren't sure what the "slow" button did.

Well, the train started up and headed out of the station just as usual. Pete couldn't tell any difference, except that the train seemed to be going slower, which seemed appropriate. So he called me on the cell phone, and I drove out alongside the train, and sure enough, it was going only 50 m.p.h. Pete continued along this way for 20 miles, and the train held steady. Then I ordered him to hit the "resume button", and the train sped back up to 60 m.p.h. for the next 20 miles.

By this time, though, we realized that Pete was going to be late getting into the station in Sugar, so I told him to hit the "on time" button. And the train did just what it said: it sped up, and pulled into Sugar exactly one hour after it left Shoe Chew. It was on that last leg that Officer Kovalevskia clocked it; I had turned around to drive back home.

Pete said that he didn't touch any buttons after I left him, so I have to believe that the F.A.R. train traveled at a steady speed. So here's where I think we can prove Kovalevskia wrong: the train traveled 60 miles in one hour, and that's an average speed of 60 m.p.h., right? That's just how speed works; if you go 60 miles in one hour, you average 60 miles per hour. It shouldn't take a rocket scientist to figure that out.

Now, Pete went 50 m.p.h. for one third of the way, and 60 m.p.h. for another third of the way; since the average of 50, 60, 70 is 60, doesn't it make sense to say that he went 70 m.p.h. for the last third of the route? If he had gone 75 miles per hour, wouldn't he have pulled into the station several minutes early? But he didn't get there early; there were witnesses in the station who can swear he was right on time.

I'm not a math whiz; perhaps you could confirm that my calculation are correct and that Officer Kovalevskia is in error. Our court date is September 26, so please respond to me by then.

Yours truly,

G. Olson Overby-Fitzpatrick

G. Olson Overby-Fitzpatrick

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