Thread: Optimization and Differentials

Hi...I'm working on a problem for the minimization of materials for a can. This juice can should hold 163 mL, and I'm already stuck on the first step. When setting up the minimization equation I should be able to graph it and get an idea of about where the minimum should be. I've got 2*Pi*r^2 + 326/r so far, but the graph doesn't show anything that looks like a zero slope anywhere - it just looks like a standard sqrt function. So before I go further with this, what am I doing wrong?

Originally Posted by marqade

Hi...I'm working on a problem for the minimization of materials for a can. This juice can should hold 163 mL, and I'm already stuck on the first step. When setting up the minimization equation I should be able to graph it and get an idea of about where the minimum should be. I've got 2*Pi*r^2 + 326/r so far, but the graph doesn't show anything that looks like a zero slope anywhere - it just looks like a standard sqrt function. So before I go further with this, what am I doing wrong?

Did you start with

and ?

I started with 163 = Pi*r^2*h;
A = 2*Pi*r^2 + 2*Pi*r*h;
h = 163/(Pi*r^2)

Originally Posted by marqade

I started with 163 = Pi*r^2*h;
A = 2*Pi*r^2 + 2*Pi*r*h;
h = 163/(Pi*r^2)

Thats correct...I just factored by suface area

Now imput that into yoru surface area to get



This means that

Now solve

If we cut this same can from hexagons instead, thereby saving even more material, could I write the area as a function of the radius as such: A(r) = 6*(1/2)(2/r)(r)? because 1/2 b*h = A; b = 2/r and the height = radius. Problem with this is that my variable cancel themselves out and I'm not sure what to do with that. Do I need to be using trig formulas?
x = (1/r*sin90)/(sin(45)) = sin(45)/r = (sqrt(2))/2r.