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  "body": "\n## Problem\n\nLet $\\ell \u003e 0$.\nConsider [[plane curves|EWS4WFTA7W]] of length $\\ell$ that start and end on a fixed straight line and otherwise lie on one side of it.\nWhich of them encloses, together with the line, the largest area?\n\nIn coordinates, the line is the $x$-axis and the curve is a piecewise $C^1$ map $t \\mapsto (x(t), y(t))$, $0 \\le t \\le 1$, with $y(0) = y(1) = 0$ and $y \\ge 0$.\nWhen the curve has no self-crossings, the enclosed area is $\\abs{\\int_0^1 y(t) \\, x'(t) \\, dt}$.\n\n## Remarks\n\nThe answer is a semicircle of length $\\ell$, with its diameter on the line; it encloses the area $\\ell^2 / (2\\pi)$.\nReflecting a curve in the line turns the problem into the classical isoperimetric problem for closed curves of length $2\\ell$.\nSo the answer follows from the [[isoperimetric inequality|KBX8RBZ626]] in the plane.\nWritten with such a parametrised curve, it is an [[isoperimetric problem of the calculus of variations|6FY0KHMK5A]].\nIts Euler–Lagrange equation with a multiplier says that the curve has constant curvature, so it is an arc of a circle.\nThe name comes from the legend of the founding of Carthage.\nDido was granted as much land as an ox hide could enclose; she cut the hide into a thin strip and laid it out against the coast.\n\n## Explanations\n\n### L1 human\n\nYou have a rope of fixed length and a straight coastline.\nYou want to fence off the largest piece of land, using the coast as one side.\nThe best shape is a half disc, with the rope as the round edge.\n",
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