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To find the volume of such a box, multiply the width by the length by the height.
The width, after cutting out the corner squares of x by x feet, will be (6-2x) feet and the length will be (8-2x) feet, and, of course, the height of the box will be x feet. So, the volume is expressed by:
Expand this to get:
This represents the volume of the box expressed in terms of x.
The graph of this cubic function looks like:
The valid range of x is x = 0 to x = 3
Using the graph to find the value of x that will produce the maximum volume is a little more than x = 1.
The actual value can be found with a little elementary differential calculus, because it's a matter of finding the "relative" maximum of the cubic curve.
Take the first derivative of the function and set it equal to zero, then solve for x.
Set this equal to zero.
The roots are:
x = 3.535... Ignore this solution as x is too large.
x = 1.131... This is the approximate value of x that will produce the largest volume.