A farmer in Devon wants a rectangular pen of area 800 m² against the straight wall of a barn…

Step 1. One variable. Width $x$, length $y = \dfrac{800}{x}$. The fence is $L = 2x + y = 2x + \dfrac{800}{x}$, for $x > 0$.

Step 2. The derivative. $L' = 2 - \dfrac{800}{x^2} = 0$ gives $x^2 = 400$, so $x = 20\ \text{m}$.

Step 3. The check. $L'' = \dfrac{1600}{x^3} > 0$, so it is a minimum.

Step 4. The length. $y = \dfrac{800}{20} = 40\ \text{m}$.

Step 5. The cost. $L = 40 + 40 = 80\ \text{m}$ of fence, and $80 \times 12 = 960$ pounds.