Expert: Jack Cheng - 11/11/2006

Question
Hello.  I am a calculus student at the University of Denver, and I have just been assigned a little project called the "Salami Curve".  My problem with this project is that I have no idea where to start.  If you don't know about this project I could explain it to you, but I think the actual instructions will make more sense (see http://www.cord.edu/faculty/lenarz/Math121/handouts/Project.pdf).  If you could help me out in this matter that would be awsome.

Answer
Hi Michael,

Thank you for a spectacular question. I REALLY liked it.
Here are my hints on how to solve it:

1) Draw a picture of the situation. It really helps to understand the question.

I drew the field as a rectangle with the long side horizontal and short side vertical (with the goal in the middle of the line), and I'll reference it as such.

2) With y representing the horizontal distance, and x representing the vertical distance to the closest goal post. Draw lines from a point on a horizontal line to the two goal posts; that creates a angle, and your objective is to maximize that angle with respect to y for each x (in other words, x is like a constant for a specific Salami point, and you have to tweak y so that the angle for that point is the biggest).

(From what I understand, you really don't have to worry about any x within the goal posts, because the closer y is, the bigger the angle.)

3) To maximize the angle, you have to find a formula for it first. The instructions give you a hint on this: it's a the difference of two inverse trig functions. I'll leave the formula for you to derive; just keep the hint from the instruction in mind (especially the fact that there are two inverse trig functions).

Now, do the maximization on the formula. The algebra is a little bit complex, but you should be able to handle it.

[Hint: you are differentiating with respect to y, and y only, so when you do the differentiation, do NOT to treat x and g (the width of the field goal) as variables. Also, remember to use the chain rule. These two things really got me several times when doing the differentiation, so be careful.]

4) After you do the maximization, you should have a formula for y in terms of x and g. The graph of that formula should be the Salami curve... but, how it applies to the field is slightly different (e.g. where on the field do you treat as the origin of the curve, and does the whole curve apply?). I'll leave those questions for you figure out.

I hope this helps, and ask again if you need any more help with the problem.

Having fun with math,
Jack

P.S. I know that my hints are little bit hard to understand without a picture. So, as you are reading, try to visual what I am saying or draw a picture. I think that would help a lot.