a rectangular sheet of cardboard 4m by 2m is used to make an open box by cutting squares of equal size from the four corners and folding up the sides.what size squares should be cut to obtain the largest possible volume?

Answer 1

let each cut of be x m
so the base is 4-2x by 2-2x
and the height is x
V = x(4-2x)(2-2x)
= 8x -12x^2 + 4x^3
d(V)/dx = 8 - 24x + 12x^2 = 0 for a max V
3x^2 - 6x + 2 = 0
x = (6 ± √12)/6
= .42 or 1.57 , but the latter would make the width negative so we will reject it

the cut-out squares should each be .42m by .42 m
or 42 by 42 cm