I have a piece of cardboard that is 7 units wide and 6 units

I have a piece of cardboard that is 7 units wide and 6 units long. I would like to use this cardboard to build a box. I will cut squares out of each corner, then fold up the flaps to create my box. My box will not have a top. My desire is to have a box that can hold the maximum amount of goods.

Volume = (7 - 2x) * (6 - 2x) * x

What will be the Maximum Volume of my box?

Solution

Corner of size x are cut from each side.

length = 7 -x -x = 7-2x

width = 6- x -x = 6-2x

height = x

Volume(V) = x(7-2x)(6-2x)

Maximum volume dV/dx = (7-2x)(6-2x) + x(-2)(7-2x) + x(-2)(6-2x)

dV/dx =0

solve we get x= 13/6 - sqrt(43)/6

= 1.073 units

Max Volume occurs at (x= 1.073 )= 1.073(7 -2*1.073)(6 -2*1.073)

=1.073*4.854*3.854

= 20.07 cubic units