Let be the length of the rectangle; the width verifies the relation: . The area of the rectangle is given by: .

Then: and if, and only if, .

As is a quadratic polynomial with negative main coefficient, the function has a maximum at the point . replacing in the definition of , we have: .

In order to have maximal area with a fixed perimeter, the rectangle should be a square.

Denote by the side length of the small squares cut off at the corners of the plate. This provides a box whose volume is given by the function defined as follows:

Now we have:

and

or |

If , nothing is left and no box is built. Therefore .

Apply the second derivative test:

and . As this number is negative, we have a confirmation that for the volume of the box is maximal.

the maximal volume is given by:

cm |

Noah Dana-Picard 2007-12-28