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Thank you for taking my question, it is very tricky!

A rectangular plot of farmland will be bounded on one side by a river and on the three sides by a single strand electric fence. With 500 meters of fence at your disposal, what is the largest area you can enclose, and what are the dimensions?

Hello Clark,

Let x=the width (perpendicular to the river), and

let y=the length (parallel to the river).

Thus, we wish to maximize the area, A=xy, subject to the perimeter, P=2x+y=500.

Solving the perimeter for y, we get y=500-2x.

Now, substitute that into the area equation: A=x(500-2x)=500x-2x^2

This can be maximized using calculus, or simply by realizing that this is a quadratic

and is maximized on its axis of symmetry, x=-b/(2a) = -500/(-4)=125

Thus, the area is maximized when x=125 and y=250...giving an area of 125*250 sq. ft. = 31,250 sq. ft.

Abe

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Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

Over 15 years teaching at the college level.**Organizations**

NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.**Education/Credentials**

B.S. in Mathematics from Rensselaer Polytechnic Institute

M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook