If 30,000 cm2 of material is available to make a box with a square base and an open top, what is the largest possible volume (in cm3) of the box?

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.