Math Help - How to find domain (optimization problem)

1. How to find domain (optimization problem)

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 $V(x) = x(10-2x)(16-2x)$ and the crit point that gives a maximum is x = 2 (which equals $144 cm^3$ ). The part I'm having trouble with is finding the domain which keeps both the length and volume positive.

2. Re: How to find domain (optimization problem)

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.

3. Re: How to find domain (optimization problem)

I see. I guess my problem was that I have trouble visualizing negative volume, or what would be outside the domain. Is the domain for this problem essentially where the box could no longer be assembled?

4. Re: How to find domain (optimization problem)

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.