Suppose you have a piece of cardboard 32 inches by 48 inches and cut equal sized squares from each corner and fold up the sides of the cardboard to form a box with an open top. What length of x maximizes the volume of the box?

Volume = lwh

l = 48 - 2x

w = 32 - 2x

h = x

V= x(48 - 2x)(32 - 2x)

V= x(1536 - 96x - 64x + 4x^2)

V = 1536x - 160x^2 + 4x^3

V' = 1536 - 320x + 12x^2

set V' = 0 and solve for x.

From quadratic formula, x = 6.28 inches

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