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Question
A farmer has 100 m of mobile fence to build an enclosure. He hesitates between a square enclosure and a circular enclosure. What shape of enclosure should he build to enclose the maximum area?

The 100 m length corresponds to the perimeter of the enclosure, P.

P_circle = 2πr = 100 ⇒ r = 100/(2π)
P_square = 4･r = 100 ⇒ r = 25

Areas:

A_circle = πr^2 = 10000/(4π) = 796 m^2
A_square = r^2 = (25)^2 = 625 m^2

So the circle has a larger area for the same perimeter.
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#### randy patton

##### Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

##### Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

Publications
J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

Education/Credentials
M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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Also an Expert in Oceanography