Calc Optimization

[SnoringFrog]

Calc has been killing me this semester, but optimization is making the rest of it look like a cakewalk. I've put 7 hours into this assignment of 7 problems (and this is after the many hints from our prof. before he let us have another go at the same assignment) and have gotten almost nowhere. I get the concept of optimization; I just can't make it work in most of the problems.

Here's what I'm stuck on (the first two I kinda have an idea of what's going on):

1. A piece of wire 12 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum (b) a minimum


I know the optimization equation is the total of the areas of the triangle and the square. And via some google-fu I found an example that based the sides of the triangle off of the sides of the square, so using that I came up with the sides of the square each equaling X and the sides of the triangle each equaling (12-4x)/3. From there I tried to take the derivitive and find the critical values and all that that you're supposed to do, but still never got to a final answer that felt correct at all. My answer for the max was x=3 (which leads to no triangle?) and for the minimum it was really weird.

2. Solve #1 if one piece was bent into a square and the other was bent into a circle

For this, I made the radius of the circle X and the sides of the squaree (12-X)/4. Again, I felt mostly lost workign though this one.

3. If a resistor of R ohms is connected across a battery of E volts with internal resistance r ohms, then the power (in watts) in the external resistor is P=(E^2 R)/〖(R+r)〗^2 . If E and r are fixed but R varies, what is the maximum value of the power.

According to my prof., this one was "pretty straightforward". From what he told us, I think I should be able to just take the derivitive of P and work from there, but when I did that I went through 2 pages of work and never arrived at an answer. That's when I gave up and moved on.

4. A boat leaves a dock at 2:00 pm and travels due south at a speed of 15 km/hr. Another boat has been heading due east at a rate at a rate of 10 km/hr and reaches the same dock at 3:00 pm. At what time were the two boats closest together?

For this one, I drew a right triangle (with the right angle in the top-left). Heigh is X and width is Y. Hypotaneuse is D. With what the professor gave us, he labeled side X with 15t (t for time) and labeled side Y with 10-10t (and a portion beyond Y with 10t). My optimization equation is D = sqrt(x^2 + y^2). I was never able to figure out a constraint. I tried working the problem anyways and ended up coming down to x = -y and writing 2pm as my answer.

I'm really really sick of this assignment at this point. I won't be able to actually fix any of my errors in time for class and still sleep enough (haven't been able to wake up on time lately), but I really want to just figure this crap out so I can get the optimization problems that will be on the test.