Question 258993
An open display box is to be made in the shape of a rectangular prism is to be made from a 3ft x 8ft piece of lumber by cutting equal squares from each of the four corners and turning up the sides. Find the volume of the largest box that can be made.
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Draw a picture of the 3' by 8' rectangular piece.
Sketch a square in each of the 4 corners that is x by x.
Imagine cutting out the squares and folding up the sides.
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Volume = (base)(height)
Base = (3-2x)(8-2x)
height = x
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Volume = (24 -6x -16x + 4x^2)(x)
V(x) = 4x^3 - 22x^2 + 24x
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To find the maximum determine the derivative:
V'(x) = 12x^2 - 44x + 24
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Solve 12x^2 - 44x + 24 = 0
3x^2 - 11x + 6 = 0
3x^2-9x-2x+6 = 0

3x(x-3)-2(x-3) = 0
(x-3)(3x-2) = 0
x = 3 or x = 2/3
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If x = 3 the volume is zero because the dimension (a minimum)
If x = 2/3 the volume is a maximum. 
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V(x) = 4x^3 - 22x^2 + 24x
V(2/3) = 7.4074 cu ft.
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Cheers,
Stan H.