Answer:
The largest possible volume of the box is 2000000 cubic meters.
Step-by-step explanation:
The volume (
), in cubic centimeters, and surface area (
), in square centimeters, of the box with a square base are described below:
(1)
(2)
Where:
- Side length of the base, in centimeters.
- Height of the box, in centimeters.
By (2), we clear
within the formula:

And we apply in (1) and simplify the resulting expression:


(3)
Then, we find the first and second derivatives of this expression:
(4)
(5)
If
and
, then we find the critical value of the side length of the base is:



Then, we evaluate this result in the expression of the second derivative:

By Second Derivative Test, we conclude that critical value leads to an absolute maximum. The maximum possible volume of the box is:


The largest possible volume of the box is 2000000 cubic meters.