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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,

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

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