The product of 2 divided by the number h and 8 more than the number k
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When the surrounding flaps are folded up, the base of the box will have dimensions by
, and the box will have a height of
. So the box has volume, as a function of
,
I don't know what technology is available to you, but we can determine an exact value for that maximizes the volume by using calculus.
Differentiating with respect to
gives
and setting this equal to 0 gives two critical points at
For the larger critical point we would get a negative volume, so we ignore that one. Then the largest volume would be about 168.5 cubic in.