Question 969740
A square is the rectangle that minimizes the perimeter for a given area.  Here is why without proving (first).  If a rectangle has area 225 m^2, 15^2 (perimeter 4*15m=60m) is the best.  For 25*9=225 , the perimeter is 68.  For 45 *5=225, The perimeter is 100.  The proof is much easier with calculus, but here is how I would go about it.

Let x and y be the length and the width, respectively.
Then 2x + 2y =perimeter; the area is xy

2x= perimeter-2y
xy,  or the area,  now become (perimeter-2y) y  =perimeter*y-2y^2
This is a parabola, with the vertex at -b/2a  or   -(perimeter)/-2 or half of the perimeter.  
So y in the area formula is half the perimeter.  That means x has to be half the perimeter as well.
I don't see the gate as adding or subtracting from the 60 m perimeter.

There are 4 corner poles.  There are 9 poles between each corner, which would be 36 poles in between..

x=y  and x^2=225 m^2, so x and y are both 15 m.  

The length of each side is 15 m

Fencing needed = 60 m.  (including 1.5 m  for the gate.  I think that needs to be mentioned)

Fencing poles needed=40

Cost is 60*35 for fencing=$2100.  60 meters *$35/meter.
Cost of poles is 40*$9.50=$380
I get $2480 MINUS the cost of the fence, which is $35*1.5 meters or $52.50
Cost now is $2422.50.

I am hoping there are no tricks with the poles and the gate.  

By the way, the maximum product for a given pair of numbers with a constant sum is when the numbers are the same.  It is sort of the same idea.  For a given amount of fence, you maximize the area it encloses when you make it a square.  Here, you have a given area, so you are looking at minimizing the amount of fence.