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  

answer: yard ramps with a level off platform are typically 36 feet long. truck, rather than at an angle, enabling access to those pallets at the very back of the truck. if you are using your yard ramp to convert a loading dock into a drive-in door, we even at 60 inches (5 feet) high and a 30 foot ramp, the slope of the ramp will be less

step-by-step explanation: