What are the dimensions of a rectangle of a fixed perimeter P that result in the largest area?
NOTE: What exactly is a fixed perimeter?
x + x + x + x, where x is the length of a side,
Or similarly, 4x = P, where P is the perimeter
P/4 = x, which is a side of the square
Thus, the dimensions that result in the largest area has to be P/4 by P/4.
to demonstrate what a fixed perimeter means I've attached a digram which shows rectangles which have all the perimeter 16 cm. As you may see the size of those rectangles differs a lot. You'll find the rectangle with the greatest size in the middle of the triangle. I've drawn this square with some thicker perimeter.
Hope this helps a little bit further.