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# Word Problems/Help

Question
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

Word Problems

Volunteer

#### Abe Mantell

##### Expertise

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!

##### Experience

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