I would observe that you need x to be greater than zero and less than half the length of the shorter side of the rectangle.
I have this optimization problem which is mostly solved: "A box is to be made out of a 10 cm by 16 cm piece of cardboard. Squares of side length cm will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top." The volume must be maximized.
The objective function is and the crit point that gives a maximum is x = 2 (which equals ). The part I'm having trouble with is finding the domain which keeps both the length and volume positive.
Yes, while the function for volume obtained is defined for all real x, we must consider the application where we cannot have a negative x and we cannot take away squares from the sheet whose side length is greater than half the length of the shorter sides. We can also see that if x is zero or the aforementioned maximal length, then the volume will be zero.