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Fixed (translation only) Isoperimetric Polyominoes. Perimeter 10. Perimeter 12. 712 can be decomposed into sums of squares in many ways ranging from the minimum of two squares up to the maximum known number of squares which can be made with the set. At present this is 25 as shown in the bottom figure below.

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Maximum Area with Fixed Perimeter An interesting geometric problem that leads to a quadratic function is that of maximizing the area enclosed by a rectangle with a given perimeter. Specifically, suppose you were given 100ft of fence and were asked to build a rectangular enclosure with maximum area.

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Triangle OFB is a part of triangle OGC. Hence, the area of OFB is smaller. The area of AFG is the area of rectangle AFOC plus the area of triangle OGC. The area of ABC is the area of rectangle AFOC plus the area of triangle OFB. Therefore, the area of ABC is smaller. The case of line DE is considered similarly.

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The area of the rectangle is the product of the dimensions. We can use the expression for we just found to write it as. This is a parabola, opening downwards, so the maximum will be its vertex, whose -coordinate we can find using. And we deduce. We just found out a very well known theorem: given a fixed perimeter, the rectangle with the maximum ...

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The result you need is that for a rectangle with a given perimeter the square has the largest area. So with a perimeter of 28 feet, you can form a square with sides of 7 feet and area of 49 square feet. This follows since given a positive number A with x y = A the sum x + y is smallest when x = y = A.

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The Figure of Maximum Area and Given Perimeter Date: 06/02/98 at 16:42:11 From: Brad Morris Subject: Calculus/geometry max/min problem Hi Math Doctor! I am a math teacher at Friends Central School in Wynnewood PA. In our Geometry book (USCMP) it states that the geometric figure of maximum area and given perimeter is a circle.