# 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?

## Expert Answers Two adjacent rectangular corrals are to be enclosed with 200 feet of fencing. Let us assume that the dimensions of the two rectangles are the same.

If the dimensions are w and l, the total perimeter of the two rectangular corrals is 4w+4l but with a common side this is equal to 4w + 3l.

Now the enclosed area is 2*w*l and as the length of the fencing is 200 ft, we have l + 2w + 2w + l + l = 200

3l + 4w = 200

`l = (200 - 4w)/3`

To maximize the enclosed area, we need to maximize: `A = 2*w*l = 2*w*(200 - 4w)/3`

`A = (400w - 8w^2)/3`

Using calculus, equate A' = 0 and solve for w:

`A' = ((400w - 8w^2)/3)'`

`= 400/3 - (16/3)*w`

`400/3 - (16/3)*w = 0`

`=> 400 - 16w = 0`

`=> w = 400/16`

`=> w = 25`

`l = (200 - 4w)/3 = 100/3`

The maximum enclosed area is 5000/3 square feet.

Denote the length of the shared side of the corrals by `l` and the width of the both corrals together as `w` (please see the attached image). Then, the required amount of fencing will be

`P = l +...

(The entire section contains 3 answers and 519 words.)

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