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:
If , nothing is left and no box is built. Therefore .
Apply the second derivative test:
the maximal volume is given by:
Noah Dana-Picard 2007-12-28