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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 https://webwork2.asu.edu/webwork2_fi...dd0b8b8e91.png 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 $\displaystyle V(x) = x(10-2x)(16-2x) $ and the crit point that gives a maximum is x = 2 (which equals $\displaystyle 144 cm^3 $ ). The part I'm having trouble with is finding the domain which keeps both the length and volume positive.

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.

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?

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.