An open box is to be made by cutting a square from each corner of a -foot-by- -foot piece $ ) of cardboard and then folding up the sides. What size square should be cut from each corner in order to produce a box of maximum volume? ( foot in)

x = L/6 is the side of the square at the corner

Step-by-step explanation:

Let´s call "x" the side of the square at the edge, then if the cardboard is  L*L the volume of the open box is:

V = ( L - 2*x ) * ( L - 2*x) * x

V(x) = [L² - 2*L*x -  2*L*x +4*x²]*x

V(x) = L²*x - 4*L*x² + 4*x³

Taking derivatives on both sides of the equation we get:

V´(x) = L² - 8*L*x + 12* x²

V´(x) = 0         L² - 8*L*x + 12* x² = 0

Solving for x

12*x² - 8*L*x + L² = 0

x₁,₂  = 8*L ±√ 64*L² - 48*L² / 24

x₁,₂ = 8*L ±√16*L² / 24

x₁,₂ = 8*L ± 4*L / 24

x₁ = 12*L /24     then e dismiss x₁ is not a feasible solution

x₂ = 4*L/24

x₂ = L/ 6 = x