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  

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1