Find the dimensions of the rectangle that will give the minimum perimeter. a farmer wants to make a rectangular paddock with an area of $ 4000 m^2$ However, fencing costs are high and she wants the paddock to have a minimum perimeter.
I have found the perimeter:
$$x\cdot y = 4000\\
    y = \frac{4000}{x}$$
$$\begin{align}\text{Perimeter} &= 2x + 2y\\
              &= 2x + 2(4000/x)\\
              &= 2x + (8000/x)\end{align}$$
How do I find the dimensions that will give the minimum perimeter?

To minimise $2(x+y)$ subject to $xy=A$ note that: $$(x+y)^2=4xy+(x-y)^2=4A+(x-y)^2$$
Now since $x$ and $y$ are both positive, the minimum value of $2(x+y)$ occurs when $4(x+y)^2$ is a minimum and hence when $(x+y)^2$ is a minimum.
Since $(x-y)^2$ is non-negative, the minimum occurs when $x-y=0$ ie when you have a square and $x+y=2\sqrt A$ and $2(x+y)=4\sqrt A$
