# math

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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.

• math -

volume= l*w*h

2h+w=35 or w=35-2h
2h+l=35 or l= 35-2h

volume= (35-2h)(35-2h)h

dV/dh= 2(35-2h)(-2)h+ (35-2h)^2 =0

4h=35-2h
6h=35
solve for h, then the cut squares are hxh.

• math -

hen the corners of size x are cut out, the dimensions of the box are

35-2x and 35-2x and x, and the volume is thus
v = (35-2x)(35-2x)x
= 1225x - 140x^2 + 4x^3

dv/dx = 1225 -280x + 12x^2
dv/dx = 0 when x = 5.83333

• calculus -

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?

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