# How do you find the optimal box size when given only the amount of material used?

I'm learning calculus 1 and I've gotten to the section on optimization. All of the questions that involve finding the optimal volume of a box set one of the dimensions to be defined in terms of another (for example they might say length and width of the Box are equal.) I understand it's probably beyond the scope of my current knowledge but I was wondering how one actually would find the optimal box dimensions to maximize volume without these restrictions.

• I could give an example: a box must have a volume of 32000 cm^3. Find the dimensions of the box that minimize the amount of material used Dec 3, 2017 at 8:58

let $$a,b,c$$ are the side lenths of the given box, then we have $$abc=32000cm^3$$ and the material that we need is given by $$M=2(ab+bc+ac)$$ can you finish? applying $$AM-GM$$ inequality we get $$2(ab+bc+ac)\geq 6\sqrt[3]{32000cm^3}$$ the equal sign holds if $$a=b=c$$
• do you know the $AM-GM$ inequalitiy? Dec 3, 2017 at 9:12
• it is $$\frac{x+y+z}{3}\geq \sqrt[3]{xyz}$$ if $$x,y,z\geq 0$$ Dec 3, 2017 at 9:14