Find the minimum length of fencing to enclose a pasture of fixed area

We want to enclose a fixed area $A$ in a rectangular pasture. What is the minimal amount of fencing needed if one side of the pasture is enclosed by a stone wall. (So, we want to minimize the length of the three other sides of the enclosure, subject to the constraint that the pasture must have area $A$ .)

Let $x$ be the length of the side of the pasture parallel to the stone wall and $y$ be the length of the sides perpendicular to the stone wall. Then, $A = xy$ is fixed, so $y = \frac{A}{x}$ . The function we want to minimize is $P = x + 2y = x + \frac{2A}{x}$ . Taking the derivative we have,

$P'(x) = 1 - \frac{2A}{x^2}.$

Thus, $P'(x) = 0$ when $x = \sqrt{2A}$ and we have

$\begin{align*} P'(x) &< 0 & \text{when} && x &< \sqrt{2A} \\ P'(x) &> 0 & \text{when} && x &> \sqrt{2A}. \end{align*}$

Therefore, $P(x)$ is decreasing when $x < \sqrt{2A}$ and increasing when $x > \sqrt{2A}$ . Hence, $P(x)$ has a minimum when $x = \sqrt{2A}$ . Furthermore,

$y = \frac{A}{x} \quad \implies \quad y = \frac{A}{\sqrt{2A}} = \frac{\sqrt{2A}}{2}.$

Stumbling Robot

Find the minimum length of fencing to enclose a pasture of fixed area

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