Question 969740: Jack would like to erect a fence on all four sides of his new garden with an area of 225m^2. The fencing will cost him $35 per metre. In order to use the least fencing he would like the garden to be a minimum. The fence will be supported by wooden fence poles, approximately 1.5 m apart and a gate 1.5 m long, made of the same fence can be placed between any two poles. Each wooden fence pole would cost $9.50
1. Calculate the lenght of each side of the garden
2.How many metres of fencing will he need?
3. How many fence poles will he need?
4. How much will the fencing and the fence poles, excluding the gate, cost him?
(Please I really need help)
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! 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.
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