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?

1 Answer

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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:


Plug-in the given perimeter of the fence.


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.


Then, express the right side as one variable. To do so, substitute EQ1.




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

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


`A'= 100-3x`

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




Next, substitute the value of x to EQ1.



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.