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