Area of a Rectangle given Perimeter? - Answers

OK: DO NOT DELETE THIS NEXT BITWikiAnswers is limited in it's asking ability, so here's the question:A farmer has 3600 feet of Fencing which he wants to use to enclose a rectangular corral and then divide it into 10 smaller, equally-sized corrals with fences parallel to the sides (See Below). Find the largest area that can be enclosed.y| and x__ _ _ _ _|_|_|_|_|_||_|_|_|_|_|I know that 3x+6y=3600 and that x*y=a but can't go beyond there...HELP!!!(END QUESTION HERE)================================(BEGIN ANSWER HERE)Referring to your drawing, the greatest area is enclosed by making the corral[ 1/6 Fence ] = 600-ft wide by [ 1/12 Fence ] = 300-ft high.Horizontal fence = 3 runs @ 600-ft = 1,800-ftVertical fence = 6 runs @ 300-ft = 1,800-ftTotal fence = 3,600 ftTotal area = (600 x 300) = 180,000 sq ft = 4.132 acres (rounded)Each pen = 120-ft wide x 150-ft high = 18,000 sq ft = 0.4132 acre (rounded)I got this in the normal way of using calculus ... setting equal to zero the derivativeof area (in terms of length of fence) with respect to one dimension ... but I don'thave any other way to prove that it's the greatest area. I did notice somethinginteresting, though. This doesn't prove anything, but it feels like maybe I'm holdinga secret in my hand that I can't read:-- We know that for a fixed perimeter, the greatest possible area with straight sidesis a square. If the perimeter is 3600-ft ... (all the fence you have) ... then the squarewould have sides of 900-ft, and the area would be (900 x 900) = 810,000 sq ft,without sub-dividing.Now stay with me here:-- Instead of two horizontal runs of fence for a single area, we're using three ... 1.5 times as much.-- Instead of two vertical runs of fence for a single area, we're using six ... 3 times as much.-- (1.5 x 3) = 4.5 .-- We could have had 810,000 sq ft without sub-dividing. Divide that by 4.5and you get 180,000 ... just what we wound up with after sub-dividing.Does this mean anything ? I don't know. But take your 180,000 sq ft anduse it in good health. That's my answer and I'm sticking to it.