SOLUTION: Find the dimensions of a rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 100 feet of fencing

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Question 1188528: Find the dimensions of a rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 100 feet of fencing
Answer by greenestamps(13200) (Show Source):
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Let x be the length of the section of fence that divides the corral in two; then there are three sections of fence of that length. The dimensions of the rectangle are then x and (100-3x)/2.

Use calculus to find the value of x that maximizes the area:

The other dimension is then

ANSWER: The dimensions of the corral are 25 feet by 50/3 feet, or 25 by 16 2/3 feet.

NOTE: This answer means that 50 feet of fencing is used for the sections of fencing in each direction.

It turns out that this is a general result for this kind of problem:

If a rectangular field is divided into several sections with fences parallel to one of the sides, the maximum area is obtained if half of the fencing is used for the fencing in each direction.

This can be proved using the method above.

Let F be the amount of fencing available, and let the rectangular corral be divided into (n-1) parts. There are then n sections of fence in one direction and 2 sections of fence in the other direction.

Let x be the length of each of the n sections of fence; then the length of each section of fence in the other direction is (F-nx)/2. The total area is then

That derivative is zero when nx=F/2 -- i.e., when the total amount of fencing for the n equal lengths is half of the total.