A country's postal service will accept a package if its length plus its girth (the distance all the way around) does not exceed 84 inches. Find the dimensions and volume of the largest package with a square base that can be mailed.

Dimensions: 18.667 x 18.667 x 14 inches

Volume: 4878.22 in3

Step-by-step explanation:

If the length plus the girth does not exceed 84 inches, we have:

The box has a square base, to length = width, then we have:

The volume of the box is given by the equation:

From the first equation, we have:

Using this height in the volume equation, we have:

To find the maximum volume, we can find the value of length that makes the derivative of the volume in relation to the length equal zero:

So the width, the height and the volume of the package are: