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?
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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)`
`4000=10x+4y`
To simplify, divide both sides by its GCF which is 2.
`4000/2=(10x+4y)/2`
`2000 =5x + 2y`
Then, solve for y.
`2000-5x=2y`
`(2000-5x)/2= y`
`1000-5/2x=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,
`A=x(1000-5/2x)`
`A=1000x-5/2x^2`
Then, take the derivative of A.
`A' =1000-5x`
Set A' equal to zero and solve for x.
`0 = 1000-5x`
`5x=1000`
`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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