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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.