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We begin by stating the information given in the problem in mathematical terms. We'll call the perimeter and the length . We know, first of all, that

We may use this to substitute for at any time:

We also know that since is the length of one side of the square end, the volume of the cuboid will be

To find the maximum volume, we note that at the endpoints of the domain (where or ) and that is differentiable everywhere. Therefore, will attain a maximum at some point where .