If 900 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

Volume $=$ (include help (units))

Step 1 ：We are given that the total material available is 900 square centimeters. This material is used to make a box with a square base and an open top. The base has a side length of x and the height of the box is h. Therefore, the total area of the material is given by the equation \(A = x^2 + 4*x*h\).

Step 2 ：We are asked to find the maximum volume of the box. The volume of a box with a square base of side length x and height h is given by \(V = x^2 * h\).

Step 3 ：We can solve the area equation for h and substitute this into the volume equation to get a function for V in terms of x alone. This gives us \(V = x*(900 - x^2)/4\).

Step 4 ：To find the maximum volume, we take the derivative of the volume function, set it equal to zero, and solve for x. This gives us the critical points \(x = -10\sqrt{3}, 10\sqrt{3}\).

Step 5 ：We substitute these critical points into the volume function to find the corresponding volumes. This gives us the volumes \(V = 1500\sqrt{3}\).

Step 6 ：Comparing these volumes, we find that the maximum volume is \(V = 1500\sqrt{3}\) cubic centimeters.

Step 7 ：Final Answer: The largest possible volume of the box is \(\boxed{1500\sqrt{3}}\) cubic centimeters.