A farmer wants to create a rectangular pen in order to raise chickens. Because of the location of the pen, the fence on the north and south sides of the rectangle will cost $5 per metre to construct whereas the fence on the east and west sides will cost $20 per metre. If the farmer has $1000 to spend on the fence, write a function in standard form to represent the area of the rectangle. (hint area = (length)(width)

Question

I don't know how to create a function that will include all the details. Please Help!

Answer 1

Let y=# of meters in the north and south sides
Let y-x= # of meters in the east and west sides

'x' would be equivalent to the total difference between the sides

I multiplied $20 with y in order to find out the cost of the N & S sides and $5 with (y-x) in order to find out the cost of the E & W.

Knowing that the total cost would be equal to a $1000, I set up the equation:
(5(y-x))+(20(y))=1000

There are 2 lengths and 2 widths in all rectangles. I did not multiply 2 with the cost since 2 sides are included in the cost ('the fence on the NORTH AND SOUTH sides of the rectangle will cost $5 per metre to construct'). The total price should be equivalent to the total price of the perimeter of the pen instead of the area.

Now first solve for y
5y-5x+20y=1000
25y-5x=1000
25y=1000+5x
y=40+0.2x

Now solve for y-x
y-x=40+0.2x-x
y-x=40-0.8x

Now we find out the area
However note that y represents TWO sides, north AND south, which are assumed to be equivalent. The same applies to y-x

So we divide y by 2. As well as y-x
A=(0.5y)(0.5(y-x))
A=(0.25)(y)(y-x)
A=(0.25)(40+0.2x)(40-0.8x)
A=(0.25)(1600+24x-0.16x^2)
A=400+6x-0.04x^2

Note: as states x represents the total difference between the two numbers
There x would be equal to the 2 times (length-width).
You can rewrite the equation like this, by representing L as length and width as W
A=400+12(L-W)+0.08(L-W)^2
Then solve from there.
A=400+12L-12W+0.08(L^2-2LW+W^2)
A=400+12L-12W+0.08L^2-0.16LW+0.08W^2
A=4(0.02W^2-0.02L^2+3L-3W+100)

So that was my solution. It was quite lengthy but I hope you got the gist of it. Most of what I did was algebra, so I simply rearranged numbers, substituted representations, and multiplied/divided numbers back and forth. Anyhow, if I did this correctly, than the solution must be correct and you can represent it again in a different way if the answer is something else. The function is basically open to all kinds of modifications. There might be a chance I might be wrong, but yeah, this is my interpretation of this problem.

Answer 2

If you are good at writing can you look at my post in current questions and help me?

Answer 3

Shorter version:
side costing $20/m --- x
side costing $5/m -----y

20(2x) + 5(2y) = 1000
4x+y=100
y=100-4x

area = xy
= x(100-4x)
= 100x - 4x^2

This can be represented by a parabola, whose vertex will yield your answer.

the x of the vertex is -100/-8 = 12.5
then y = 100 - 4(12.5) = 50

The field should be 50 m by 12.5 m

check:
20(25)+5(100)=1000