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A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume $ V $ of the box as a function of x.

$240 x-64 x^{2}+4 x^{3}$

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Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Okay. As this problem describes, we start with a rectangular piece of cardboard and squares are cut from the corners. The squares have dimensions X by x, and so the remaining piece of cardboard is folded up to make a box. Okay, so here's our box. Now this dimension of the box right here that came from this side of the cardboard. And how long is that? Well, the original side length was 20 but we subtracted X over here and X over here. So that length is 20 minus two x and then this side of the box is this length over here. How long is that? While the original piece of cardboard was 12 but then we subtracted except there and ex down there. So that is 12 minus two x, and the height of the box is the height of the chunk cut out, and that is X. So now that we have descriptions for all of those dimensions, we can find the volume of the box because we know that volume is like times with times height. So the volume as a function of X, will be acts times 20 minus two X times 12 minus X. Now that's a perfectly acceptable answer. But if you want to simplify your answer, you would end up multiplying all those quantities together and we can do that. So I'm going to leave the X there for a moment, and I'm going to multiply the by no meals by using the foil method. So 20 times 12 would be to 40 and then multiply the outsides. We get minus 40 x, multiply the insides, we get minus 24 X, and then we multiply the last and we get plus four x squared. And then we can combine the like terms. We can combine the minus 40 X and the minus 24 X and then lastly, we could distribute the X. So the volume of the box is 240 x minus 64 X squared plus four x cubed. I'm not saying you would have to do that simplification, but if you wanted to, there it is