A rectangular pen is to be fenced off with 36 $m$ of wire, using the whole length of wire. What is the greatest area it can enclose, in $m^2$?

With the perimeter fixed at 36 m, $l+b=18$, and the product lb is largest when the two are equal — a square of side 9 m, giving $81{\text{m}}^2$. Any other split loses area: $10\times 8=80,12\times 6=72,15\times 3=45.(72{\text{m}}^2$ is one of those legal but smaller pens; $324{\text{m}}^2$ squares half the perimeter, $18^2$, forgetting that the side is a quarter of 36; $1296{\text{m}}^2$ squares the whole perimeter.)