Nov 29, 2008

Arccotangent

I love calculus, with all of its rules and formulas, not just for its own sake but also because it's useful in real life. However, there are some elements of the AP Calculus curriculum that I find unnecessary. For example, last week I was teaching my calc students how to find derivatives of inverse trig functions (arcsin, arctan, etc.) and I had a hard time pretending to be enthusiastic about them. Why memorize derivatives of functions that you can just find in a table in the back of a book, and that have no use in real life?

Well, a teacher of mine gave me this problem as an example of a real life application of these derivatives, and I think it's pretty neat. Can you solve it?

Miguel and his friends went to go see "Quantum of Solace" in a movie theater, and they wanted to get the best possible seats. The screen was 8 feet high, and the bottom edge of the screen was 5 feet above their heads. How far back should Miguel sit with his friends if he wants to have the best view of the screen? Assume that the theater doesn't have stadium seating, and that the floor is flat all the way back. Also assume that there are no other tall people there to block the view.





















(Hint: To find the "best" view, Miguel wants to maximize the viewing angle)

4 comments:

jd2718 said...

What angle will be best? At some point, couldn't we reach a "too big" angle, making it hard to take in the whole screen?

Maybe not. And a nice one all the same.

Good luck, writing and stuff...

Jonathan

David said...

Hey Jonathan, thanks for the encouragement!

There is a sweet spot for how far back to sit because if you go way far back, the angle gets smaller the further away you get. But if you get too close, if you sit right underneath the screen, the angle shrinks to zero. At that sweet spot, the viewing angle is less than 30 degrees, which I think is pretty comfortable.

David said...

This is the known as the problem of Regiomontanus.

David said...

Thanks for the heads-up. I'm always amazed by how much older math problems and theorems are than I would have imagined.