If 1,200 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters. answer to the nearest cubic centimeter without commas. for example, if the answer is 2,000 write 2000.

An open-top box uses a minimum of material when the shape of it is half a cube. Two such boxes would form a cube with area 2400 cm², so side area of 2400/6 = 400 cm², and edge dimensions of √(400 cm²) = 20 cm.

The volume of the half-cube is then (1/2)·(20 cm)³ = 4000 cm³.

Let x represent the edge-length of the square base of the box. Then the height of the side is v/x², and the total area of the open-top box is

... 1200 = x² +4x(v/(x²))

Multiplying by x  and subtracting the left side of the equation gives

... x³ -1200x +4v = 0

Differentiating, we have

... 3x² -1200 +4v' = 0

At the maximum volume, v' = 0, so this becomes

... 3x² -1200 = 0

... x² = 400

... x = 20 . . . . . . . . as argued above.