A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use fencing selling $5 a foot.. ...while the remaining two sides use fencing selling $2 a foot. What are the dimensions of the greatest area that can be fenced in at a cost of $4000?  

Expert Answers

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Let x and y be the sides of the rectangle.

Then, set-up the equation for the total cost of the fence. To do so, let's use a $5 per foot fencing for the two sides "x" . And, $2 per foot fencing for the two sides "y". 

So the equation is:

`4000 = 5(2x) + 2(2y)`


To simplify, divide both sides by its GCF which is 2.


`2000 =5x + 2y`

Then, solve for y.


`(2000-5x)/2= y`


Since we have to maximize the area of the rectangle, set-up the equation of area.

`A =xy`

Note that `y= 1000-5/2x` . So, 



Then, take the derivative of A.

`A' =1000-5x`

Set A' equal to zero and solve for x.

`0 = 1000-5x`


`x = 200`

Substitute the value of x to `y = 1000-5/2x` .

`y=1000-5/2*200 = 1000-500=500`

Hence, the dimensions of the rectangular plot that would maximize its area, given the total cost of fencing, is `200 xx 500` ft.

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