# Calculus project help

1. May 4, 2008

### gtcox

My class is doing an optimization problem, The shape of a can. I have attached a copy of the problem.

I have done problems 1 and 2 and now am stuck on problem 3.
What I've tried so far is to differentiate the function 4sqrt3 r^2 +2Pi rh +k(4Pir +h) with respect to r where h=v/Pir^2. I know that to optimize the function I have to set the derivative equal to zero and solve to find min/max but I have to show that when the expression is minimized it is equal to the third root of volume over the constant k. What baffles me is that all the variables V, h and r are in the minimized expression. How did that happen?

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2. May 4, 2008

### HallsofIvy

Staff Emeritus
Is V a variable? It looks to me like this problem is asking for the most efficient dimensions for a can of a fixed volume- that is V is a constant. And since you also know that $h= V/\pi r^2$, you can replace h by that and have only r as a variable.

3. May 4, 2008

### gtcox

That's what I tried, but when you substitute for h and differentiate the result is in V, r and k. When the expression is minimized it is in terms of V,r,k,and h. I just don't know how all the h got back into the equation along with V. If i set the derivative equal to 0 and solve for V, then replace V with Pi(r^2)(h), the V dissapears. If i don't solve for V then I don't know how to get the h back into the equation. Am I making any sense?