If 1000 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.

The area of the base is x^2> The height is h. Each side of the box has area xh. There are 4 sides of the box so the total surface area of the box is x^2+4xh and that is equal to 1000. Solve that equation for h: x^2+4xh = 1000 h = (1000-x^2) / 4x so the Volume = x^2[ (1000-x^2) / 4x] Simplify and get V = 250x-x^3/4 The volume will be a maximum when its first derivative is 0. V' = 250-3/4x^2 Set to 0 and solve. x=18.26 Now plug into the volume function to find the maximum volume: V=250 (18.26) - (18.26) ^3/4 V = 4564.35 - 1522.10 = 3042.25