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)
I don't know how to create a function that will include all the details. Please Help!
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.
If you are good at writing can you look at my post in current questions and help me?
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
Well, it seems like this farmer has some "fowl" play to do with the chicken pen! Don't worry, I'll help "peck" away at this problem for you.
Let's say the length of the rectangle is "x" metres, and the width is "y" metres. We know that the cost of the north and south fences is $5 per metre, so the cost for those fences is 2 times the length, or 2x.
Similarly, the cost of the east and west fences is $20 per metre, so the cost for those fences is 2 times the width, or 2y.
Given that the farmer has $1000 to spend, we can set up an equation to represent this:
2x * $5 + 2y * $20 = $1000
Simplifying this equation, we get:
10x + 40y = 1000
Now, we want to find the area of the rectangle, which is given by the formula: area = length * width, or A = xy.
So, the function in standard form to represent the area of the rectangle is:
A = xy
And the equation representing the cost of the fences is:
10x + 40y = 1000
I hope this "eggsplains" it for you! If you have any other questions, just "cluck" away!
To create a function that represents the area of the rectangle, we need to define the length and width of the rectangle based on the information given.
Let's assume the length of the rectangle is "x" metres and the width is "y" metres.
The cost of the fence on the north and south sides is $5 per metre, so the total cost for these two sides (which are of length "x") would be 2x * $5.
Similarly, the cost of the fence on the east and west sides is $20 per metre, so the total cost for these two sides (which are of length "y") would be 2y * $20.
According to the problem, the farmer has $1000 to spend on the fence. So we can write an equation for the total cost of the fence as follows:
2x * $5 + 2y * $20 = $1000
Now let's simplify this equation and rewrite it in standard form by multiplying out all the terms:
10x + 40y = 1000
Finally, we need to find the area of the rectangle, which is given by the formula: area = length * width = x * y.
Therefore, the function in standard form that represents the area of the rectangle is:
area(x, y) = x * y.
Hope this helps!