Abox (with no top) will be made by cutting squares of equal size out of the corners of a 21 inch by 48 inch rectangular piece of cardboard, then folding the side flaps up. find the maximum volume of such a box. round to the nearest cubic inch

The volume of the box as a function of the cut size (x) is

... v(x) = x(21 -2x)(48 -2x)

... = 4x³ -138x² +1008x

Setting the dirivative to zero lets us find the value of x that maximizes the volume.

... v'(x) = 12x² -276x +1008 = 0

Dividing by 12 gives ...

... x² -23x +84 = 0

and the quadratic formula tells us

... x = (23 -√(23²-4·84))/2 = 11.5 - √48.25 ≈ 4.55378

The volume for this cut dimension is

.. v(4.55378 in) ≈ 2106 in³

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Such a problem is easily solved by a graphing calculator that can tell you the extreme value of the volume function and where it is located.

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