Expert: Ahmed Salami - 8/5/2009

Question
a rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand of electric fence.  With 2200m of wire at your disposal, what is the largest area you can enclose, and what are its demensions?
a)what is the max area of the rectangular plot?=m^2
b)What is the length of teh shorter side fo teh rectangular plot=m
C)What is teh lenght of the longer side? =m

Answer
Hi Clint,
Let y be the length of the fence opposite the river and the other two sides each having length x. The sum of these sides should be equal to the total length of the electric fence i.e
x + x + y = 2200
2x + y = 2200
The area of the rectangular plot formed is then
A = xy
But from 2x + y = 2200
y = 2200 - 2x
Therefore,
A = x(2200 - 2x)
 = 2200x - 2x²
The largest area is achieved when dA/dx = 0
Now,
dA/dx = 2200 - 4x
equating to zero,
2200 - 4x = 0
2200 = 4x
x = 550m
y = 2200 - 2x
 = 2200 - 2(550)
 = 2200 - 1100
 = 1100m
And so,
a)The maximum area is
A = (550)(1100)
 = 605000m²

b)The length of the shorter side is
x = 550m

c)The length of the longer side is
y = 1100m

Regards