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.

side side side

side angle angle

angle side angle

side angle side

step-by-step explanation: