
Re: Trigonometric area optimization
Posted:
Dec 17, 2012 9:57 AM


Well, Peter S., I did not really deserve all the credit, but I'll take it and bank it, since I get so little these days. It was petty of me to bring up the subject of credit, but thought it was a good way to introduce myself and to prolong the discussion here, and, judging from earlier entries, it seems this particular discussion group is not heavily subscribed to.
My insight was realizing that the two other lines form a right triangle with vertex at the origin. Having drawn the line, I could see that what we have is a right triangle rotating at the vertex, with the hypotenus formed by the static, straight line. No great achievement, but it does show that problems do require a bit of thought, and it is good to form a picture of the problem before solving it.
I think our student might have frozen when presented with this problem, as happens to me now and then when I am put on the spot by a student who shows me a problem I cannot solve at the moment. I usually tell the student that I shall work on it and email him an answer, a promise which I dutifully carry out.
While I suspected the minimum area would be the isosceles triangle, I did take the time to "prove" it as part of the solution to the problem. Not sure if the instructor would accept a glib statement from a student such as, "And of course, as we know," etc. etc. It immediately occurred to me that the minimum area was probably the isosceles construction, but being unfamiliar with said theorem, it behooved me to work it out, both for my own edification and for the student's. I used calculus, since Adam said: "I just couldn't, for the life of me, find a suitable function which I could take the derivative of". I wanted therefore to present a function that he could take the derivative of, since that is what he asked for.
This turned out to be a rather easy and fun problem to solve. It was nice to have Adam present the problem, and to observe the immediate response by many knowledgeable voices. Glad to meet you all, and to add my 1.73 cents to this discussion group.

