October 15, 2001

Mesquite Trees, Anon.
Thorn Avenue
Lubbock, TX 79414

Dear Calculus Student,

You don't know me - my name is Betty Kant, but you do know my boss, George. He's such a sweet guy and a wonderful boss. I can't imagine anything I'd rather do for a living than measure thorns for him. He tells me that it was you who came with the new measuring technique-I have to tell you that that's made our job a whole lot easier.

But that's not actually why I'm writing you. I was hoping - since you're so good at solving our problems - that you could help me with a a problem that's been plaguing my track team.

We're all amateurs, and we all suffer from having no depth perception. Normally this is no problem. When you run events on the track, you just go where the track goes and stop when you cross the line - no big problem. In fact, we can all do the quarter-mile (well, 400 meters) in just about 66 seconds. Pretty respectable for a bunch of amateurs! The sticky part comes when we try to do relays. We're horrible at hand-offs!

The thing is, in a track relay, each runner has to hand a baton to the next runner in a short box called the passing zone, which is about 20 meters long. Let's suppose our first runner, Amanda, wants to hand the baton to me. She comes chugging down the home stretch. If I stand at the beginning of the box and wait for her, she blows by me and it takes me a while to get going. I'm allowed to start running before she gets to the passing zone, but sometimes (no depth perception) I start too soon, and then I nearly run out of the end of the box before she gets there, so I have to slow down and wait again. The same dilemma happens when I pass to Carla or when Carla passes the baton to Darlene. It's awfully embarrassing to have such lousy handoffs when we can run so respectable all alone. Darlene flat-out refuses to join the relay anymore after one particularly sloppy race a year and a half ago.

But suppose we put a yellow bandanna down on the ground at just the right distance before the passing box. Then when Amanda runs past the bandana, she could yell "go!" and I could run as hard as I could - I know I can get up to full speed in just 2 seconds if I don't have to worry about Amanda meeting up with me - and we'd have a near perfect hand-off. Then I could do the same for Carla, and Carla for Darlene.

Do you have any way of figuring out where we put the bandanna? There's a big track meet coming up on Veteran's Day, November 11. If I could get your answer by Election Day, November 6, we would have time to work it out in practice before the race. I'd really appreciate it!


Betty Kant

P.S. Could you give us an idea about:

a. How many seconds after Amanda yells "go" will we make the exchange?

b. Could you provide me a graph which plots, from the time Amanda yells "go" until we make the exchange, my position in the passing zone and Amanda's position on the track and into the passing zone (as functions of time)? I could use this as a persuasive argument on Darlene that it will all work!