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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.

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  • $\begingroup$ 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 $\endgroup$
    – ArtaSoral
    Dec 3, 2017 at 8:58

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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 $$a=b=c$$

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  • $\begingroup$ I cannot, the only way I've learnt to optimize involves taking the derivative of one variable against another, which means I have to Define all my variables in the end as one of two options $\endgroup$
    – ArtaSoral
    Dec 3, 2017 at 9:12
  • $\begingroup$ do you know the $AM-GM$ inequalitiy? $\endgroup$ Dec 3, 2017 at 9:12
  • $\begingroup$ No, I haven't heard of that. $\endgroup$
    – ArtaSoral
    Dec 3, 2017 at 9:13
  • $\begingroup$ it is $$\frac{x+y+z}{3}\geq \sqrt[3]{xyz}$$ if $$x,y,z\geq 0$$ $\endgroup$ Dec 3, 2017 at 9:14
  • $\begingroup$ I found a Wikipedia page on it I'm working at it now $\endgroup$
    – ArtaSoral
    Dec 3, 2017 at 9:14

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