A farmer wants to make three identical rectangular enclosures along a straight river, as in the diagram shown below. If he has 1200 yards of fence, and if the sides along the river need no fence, what should be the dimensions of each enclosure if the total area is to be maximized?

Answer 1

Let the length be 3x, and the width be y.
Maximum area is achieved when 3x = 4y = 600
So the dimensions are 600 by 150
To see this, note that
3x + 4y = 1200
The area is
a = 3xy = 3x(1200-3x)/4
find that da/dx = 0 when x = 200

In all these area problems, maximum area is when the available fencing is divided equally among the lengths and the widths.

Answer 2

Let,
x = length
y = width
We know that the total length fences is 1200 m
l 3x + y = 1200
l y = 1200-3x (1)
l A = xy (2)
Substitute (1) in (2)
l A = x(1200-3x)
l A = 1200x-3x2
Domain: 0 ≤ x ≤
Using derivatives,
l = 1200-6x
Optimum: l 1200-6x=0
l 6x = 1200
Answer:
x = 200m
y = 1200 - 3(200)
y = 600m