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?
Best Answer
You could just minimize the function that you found for the perimeter. The minimum is attained when the derivative is zero. So calculate: \begin{equation} \frac{d}{dx} perimeter= \frac{d}{dx}(2x+\frac{8000}{x})=2-\frac{8000}{x^2}=0. \end{equation} This gives $2=\frac{8000}{x^2}$, so $4000=x^2$. We then get \begin{equation} x=\sqrt{4000}, \end{equation} since the negative solution is not an option.