PreCalculus Help Please
Mathematics

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A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. Length of adjacent corral 1 is x, width is y. Length of the two corrals together is 2x, width y. Write the area A as a function of x.
Let x and y be the length and width of the rectangular corral.
Since the two rectangular corrals are adjacent to each other, let one side of each corral (x) share a common fence.So, their total perimeter is:
`P=3x+4y`
Plugin the given perimeter of the fence.
`200=3x+4y`
Then, isolate either of the variable. Let the y be isolated.
`(2003x)/4 =y` (Let this be EQ1).
Next, setup the equation for total area of the two rectangular corrals.
`A=2(xy)`
Then, express the right side as one variable. To do so, substitute EQ1.
`A=2x(2003x)/4`
`A=x(2003x)/2`
`A=(200x3x^2)/2`
To determine the dimensions of each rectangular corral that would maximize area, take the derivative of area.
`A'= (2006x)/2`
`A'=(2(1003x))/2`
`A'= 1003x`
Then, set A' equal to zero and solve for x.
`0=1003x`
`3x=100`
`x=100/3`
Next, substitute the value of x to EQ1.
`y=(2003x)/4=(2003(100/3))/4=(200100)/4`
`y=100/4=25`
Hence, the length and width of each rectangular corrals is `100/3` ft. and 25 ft, respectively
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