A box with an open top is to be constructed from a 4ft by 3ft rectangular piece of cardboard...

Question

A box with an open top is to be constructed from a 4ft by 3ft rectangular piece of cardboard by cutting out squares or rectangles from each of the four corners, as shown in the figure, and bending up the sides. One of the longer sides of the box is to have a double layer of cardboard, which is obtained by folding the side twice. Find the largest volume that such a box can have. 

Answer 1

Volume, V = Width•Length•Height

Width = 3-3x Because of double fold

Length = 4-2x

Height = x

V(x) = (3-3x)(4-2x)x = 6x³-18x²+12x

V'(x) = 18x²-36x+12

Divide through by 2 and set to zero to solve for x

9x²-18x+6 = 0

x = 1+√3 / 3, 1-√3 / 3

V(1+√3 / 3) = -2.309

V(1-√3 / 3) = 2.309 ft³ <--- Maximum Volume