Draw a diagram.
First, draw a rectangle representing the un-cut piece of cardboard.
Label the sides. L and W would be good names (representing the length and width, respectively).
Now draw four dotted lines, representing where the piece of cardboard will be folded up. The portion between each dotted line and its respective edge will represent a side of the box, so each dotted line needs to be offset towards the center the same amount as all of the others, and parallel to its edge, so that after the sides of the box are folded to their final (vertical) position, all sides of the box are the same height.
This offset distance is "x". Label it. Draw solid lines from each of the four dotted-line-intersections to the sides. If drawing correctly so far these represent corners that are squares of side-length "x" that would need to be cut out (waste material) before building the box, to allow the sides to be bent upward without crinkling up the material.
Figure out the new length, width and height of the rectangle. For example, if the original length was 50 inches, the new length will be x less due to the fold on one end and another x less due to the fold on the other end. 50 Less two "x" can be expressed as "(50-2x)".
Volumes of rectangular prism shapes (such as the volume of this box) are given by length times width times height.
You should be able to tell from your diagram how high the box will be after it gets folded on the dotted lines. Hint: it is not a specific number, but does represent your choice of how far in to draw the dotted "margin" fold lines from each edge
Here, the "=" represent the solid lines (edges of the box when it is ready to cut) and "." represent dotted lines (where you would fold)
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= = = =
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= . . =
= . . =
= . . =
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= = = =
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After using your diagram to identify expressions for the length, width, and height of the box in terms of "x", in other words expressions that are still true regardless of which value of x you actually chose (realizing that x must be greater than zero (or the box would have no height) and less than 13 inches (or the box would have no width), write an expression that finds l*w*h, and simplify it by multiplying these expressions together. You can either use the box method or FOIL them. Don't forget to multiply all three factors together and group like powers (simplify!)
For example, if you choose to keep using L and W until the very end : (L*W being dimensions of the original cardboard, V, l,w,h being volume and dimensions of the completed box)
V = l*w*h = (L-x)*(W-x)*x = (LW - (W+L)*x + x^2)*x = L*W*x - (W+L)*x^2 + x^3
Finally you probably ought to rewrite the polynomial in standard form (highest power first):
Keeping L and W as variables we would get:
V(L,W,x) = x^3 -(W+L)*x^2 + L*W*x
