Given the closed box with a square base, the volume V is represented by the formula V = l*w*h and the surface area SA = 2lw + 2lh + 2wh. As the base is a square, then length l equals width w so the formula for the surface area simplifies to SA = 900 = 2l^2 + 2lh, and volume simplifies to V = l^2*h.
Separate the height h from the surface area equation as h = (900 - 2l^2) / (2l) and substitute it in the volume expression to get V = l * (450 - l^2).
To identify the maximum volume, we find the derivative of the volume function, set it to zero and solve for l. This will provide us potential points (critical points) for the maximum volume. As we're seeking a maximum point, the second derivative of the volume function should be negative.
After calculating, we get two potential solutions for l where derivative is zero. Not all critical points might denote extreme values. To understand the nature of these critical points (whether they denote maximums or minimums), we take the second derivative of the volume function and substitute the critical points in there. Find the second derivative of the given volume function, substitute the values of l from the solutions obtained above into this second derivative.
We then select the length that ensures second derivative less than zero as that denotes a maximum point, substiting it back into the volume function to calculate the maximum volume.
After doing this, our calculated length l at maximum volume is approximately 12.2 meters, and the corresponding maximum volume is approximately 3674.2 cubic meters.