The Biggest Area

You have 40 metres of fence to go all the way round a rectangle. If the width is w, the two widths and two lengths add up to 40, so one width and one length add up to 20. The length is $20-w$, and the area is $A\left(w\right) = w\left(20-w\right)$.

Which width gives the most room? Multiplied out, $A\left(w\right) = -{w}^{2}+20w$: a parabola opening down, so its vertex is the greatest area. Its zeros are w = 0 and w = 20, where the rectangle is flat and has no area. The vertex is halfway between them, at w = 10, and $A\left(10\right) = 10\left(10\right)$, which is 100 square metres.

The best rectangle is a 10 by 10 square. That is no accident: for a fixed length of fence, a square always encloses the most area of all rectangles.