math
posted by Sejul .
open top rectangular box made from 35 x 35 inch piece of sheet metal by cutting out equal size squares from the corners and folding up the sides. what size squares should be removed to produce box with maximum volume.

volume= l*w*h
2h+w=35 or w=352h
2h+l=35 or l= 352h
volume= (352h)(352h)h
dV/dh= 2(352h)(2)h+ (352h)^2 =0
4h=352h
6h=35
solve for h, then the cut squares are hxh. 
hen the corners of size x are cut out, the dimensions of the box are
352x and 352x and x, and the volume is thus
v = (352x)(352x)x
= 1225x  140x^2 + 4x^3
dv/dx = 1225 280x + 12x^2
dv/dx = 0 when x = 5.83333 
an open box is to be made from a piece of metal 16 by 30 inches by cutting out squares of equal size from the corners and bending up the sides. what size should be cut out to create a box with the greatest volume? what is the maximum volume?