Maximum Area: a rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. what dimensions should be used so that the enclosed area will be a maximum?

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lemjay | High School Teacher | (Level 3) Senior Educator

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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`

Plug-in the given perimeter of the fence.

`200=3x+4y`

Then, isolate either of the variable. Let the y be isolated.

`(200-3x)/4 =y`        (Let this be EQ1).

Next, set-up 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(200-3x)/4`

`A=x(200-3x)/2`

`A=(200x-3x^2)/2`

To determine the dimensions of each rectangular corral that would  maximize area, take the derivative of area.

`A'= (200-6x)/2`

`A'=(2(100-3x))/2`

`A'= 100-3x`

Then, set A' equal to zero and solve for x.

`0=100-3x`

`3x=100`

`x=100/3`

Next, substitute the value of x to EQ1.

`y=(200-3x)/4=(200-3(100/3))/4=(200-100)/4`

`y=100/4=25`

Hence, the length and width of each rectangular corrals is `100/3` ft. and 25 ft, respectively.

 

Here is a similar problem with 400 total feet of fencing instead of 200.

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