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An open box is made from a piece of metal 40 cm by 50 cm by cutting a square from each corner and folding up the edges. The volume of the box is 1000 cm$^3$. (Recall that volume is area of the base times height.) What is the length of the sides of the squares?

Thus far I have that $A=l\times w$. Thus, $(40-2x)(50-2x)=A$ and volume is $l\times w\times h$. We get $1000=(40-2x)(50-2x)x$. Is this correct? How do I solve the cubed equation then?

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Yes that appears correct so far. As for how to solve for $x$ in the cubic polynomial, unfortunately it is not particularly clean or easy. Move everything to one side to get $4x^3-180x^2+2000x-1000=0$.

I will not go into detail on the specifics of solving a general cubic equation. The wiki page has plenty of information on the general solution for the roots of an arbitrary cubic and explains it in much more detail than I can do here. You will find that for these specific numbers there are three different zeroes to the polynomial but one will not make sense as an answer. One is near $0.5$, another is near $18$, and the third is near $27$. Note that the zero near $27$ though would require you remove more than is possible from the metal sheet since $2\cdot 27>40$.

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As JMoravitz said, the two viable solutions are about $0.5$ and $18$. You could find these approximations easily by constructing a graph of the polynomial he gives. The slope of the graph at $x=0$ is $2000$. The $y$-intercept is at $1000.$ The graph is nearly straight at this point $(0,1000)$, so the tangent line at $x=0$ crosses the $x$-axis at $0.5$ which is very near to the solution $x=0.52$.

Likewise, there is an inflection point at $x=15.$ Construct the tangent line there and see that it crosses the $x$-axis near $18$.

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