If 2200 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

Volume v = x²y; x² + 4xy = 2200; y = (2200 - x²)/4x; v = x[(2200 - x²)/4] = 550x - x³/4; v' = 550 - 3x²/4 ^{= 0;}

x = √2200//3 = 10√66/3; v'' = - 3x/2 < 0 at x = 10√66/3. So, we have maximum volume.

v_max = 10√66/3(550- 550/3) = 11000√66/9 = 9929.38