Question 229619: n open box will be made out of a piece of metal that is 24 inches by 30 inches by cutting squares out of the corners and folding up the sides. Write an equation describing the volume of the box in terms of x and find how big we should cut the squares to maximize the volume of the box.
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! Starting with the 24" X 30" piece of metal, remove the four corners by cutting out x by x squares.
The volume of the box made by folding up the sides can be expressed by:
 where x = the height of the newly-formed box, (24-2x) = the width of the base, and (30-2x) = the length of the base.
Now that we have the equation for the volume of the box , we can find its maximum value by differentating V with respect to x and setting this equal to zero.
What we are really doing, of course, is finding the points on the cubic curve at which the slope is zero, thus we are findinding the relative maximum and relative minimum points on this cubic curve.
Differentiate.
 Set equal to zero.
 Simplify by factoring a 12.
 so, by the zero product rule, we have:
 Since this won't factor, we'll use the quadratic formula to solve for x.  where a = 1, b = -18, and c = 60.
 Evaluate.
 or 
Notice that the larger solution is of no practical value since, if the cut-out squares are 13.58" by 13.58", this would take away more material than you have in the original piece of metal!
So, to obtain the maximum volume for the box, the squares must measure 4.42" by 4.42"
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