Math

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.
Volume = ?

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  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

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