A rectangular box with a square base and no top is to be made of a total of 120 cm2 of cardboard. Find the dimensions of the box of maximum volume.

The dimensions of the box of maximum volume are: Length: 6.325 centimeters, height: 3.162 centimeters.

Step-by-step explanation:

The volume (), measured in cubic centimeters, and the surface area of the rectangular box (), measured in square centimeters, with a square base are represented by the following formulas:

(1)

(2)

Where:

- Length of the side of the square base, measured in centimeters.

- Height of the box, measured in centimeters.

By (2) we clear the height:

And expand (1) by this result:

Lastly, we proceed to perform First and Second Derivative Test to find critical values associated with maximum volume:

First Derivative Test

(1)

Second Derivative Test

(2)

We conclude that critical value leads to a maximum volume.

If we know that , then dimensions of the box are, respectively:

The dimensions of the box of maximum volume are: Length: 6.325 centimeters, height: 3.162 centimeters.

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step-by-step explanation: