If 1600 square centimeters of metal is available to make a box with a square base and an open top, find the largest possible volume of the box.

Question

Volume = ?

Answer 1

side of square bottom = s
height = h

area = 1600 = s^2 + 4 s h
so h =(1600-s^2)/4s

V = s^2 h
V = s^2 (1600-s^2)/4s = (1600 s - s^3)/4

4 V = 1600 s - s^3
max/min when derivative = 0
0 = 1600 -3 s^2
s^2 = 1600/3
s = 40/sqrt 3
4 V = 1600 (40/sqrt 3 ) - 40^3/3sqrt3
4 V sqrt 3 = 1600(40) -40^3/3
4 V sqt 3 = 42666 2/3
V = 6158

Answer 2

40