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# A rectangular box without a top (a topless box) is to be made from 12 $\mathrm { ft } ^ { 2 }$ of cardboard. Find the maximum volume of such a box.

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The thing we want to maximize is the volume, hence our function to be maximized is $f(x,y,z)=xyz$, such that $x$ , $y$ and $z$ are the box dims.

And the constraint we have is the lateral area plus the base area is restricted to be 12 ft$^2$, hence $2(xz+yz)+xy=12$, such that the base dims are $x$ and $y$.

Hence our problem is to maximize $f(x,y,z)$ under the constraint $g(x,y,z)$

$f(x,y,z)=xyz;\qquad{g(x,y,z)=2(xz+yz)+xy-12=0}$

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