Dear Calculus Student,

Brilliant. You guys are brilliant! How I ever would have come up with something like that "arcsine" business, I just don't know, but it worked. I owe you guys big time.

Speaking of that, I'm really surprised to hear you didn't get your payment. It must have been a computer error, or maybe the post office lost it. I'll get my accountants on that right away, priority one. Don't worry about the money. Do NOT worry about the money. You've got my word that I'll take care of it. You know I'd do anything for you, after all you've figured out for me.

Speaking of which, it'd help me out a lot if you could help me up just a bit with hooking up an outflow pipe. Getting this taken care of would ease the cash-flow situation, if you see what I mean. Not that there's a problem, mind you, but business setbacks happen sometimes. A little patience -- a little assistance here and there -- helps to get things back on the right track.

The outflow pipe is supposed to hook up straight beneath the belly of the water tank that you're so familiar with. What we need to do, you and I, is figure out how to cut this pipe. I have 4 flat sheets of metal, each 1.6 feet long. I want to trim one end of each of the sheets and then join them together at the sides, so that I can make them into a pipe which is 2 feet wide. I'll attach the pipe to the bottom of the water tank, and if we do it right, the pipe will fit snug with no holes or gaps.

Now, I've figured that I could cut each of these 4 pieces the same. But cutting the top in a straight line doesn't work. I also tried tracing out the edge of the water tank on one of the sheets, but as soon as I curved the sheet, it didn't fit right again. So what shape should I cut them in?

I have to admit, I thought that pr² was a lot nicer than the other formula you gave me, but as I said in my first letter, I can punch buttons on the calculator with the best of them. So go ahead and do your worst.*

So, I think that'll about do it. You'll let me know about the outflow pipe, and I'll see to it that you get every last cent you're owed. You have my word. You're great, you know, you really are.

By the way, if you run into my big brother, please tell him that that thing I called him about was just a joke and that I didn't mean it the way it sounded. I'll make it up to him later.

Remember, it's not what you know; it's who you know -- and now you know me.

Sincerely,

Brent Trachte
CEO, Fowl Play, Inc.

P.S. Here are some diagrams of the way I'd like the finished outflow pipe to look when it's attached to the water tank. Hope this helps!

. . .
My Tank and Pipe: Views from the side and front

My Tank and Pipe: View from below

This problem originally comes from Haas Metal Engineering in the Topeka, Kansas area. Thanks very much to Kevin Charlwood (Washburn University) for passing this great problem along my way!