Project C10 - Isoperimetric Inequality Project C10 - Perimeter, Area, and the Isoperimetric Inequality

Suppose you are given a length of rope and you tie the ends together to form a simple closed curve. How would you arrange the rope so that the enclosed area is as large as possible? You'll find the answer to this question and several related questions as you work this project. The first thing you'll need to do is get a copy of the program Wingeom available at http://math.exeter.edu/rparris/wingeom.html.

  1. Your first task will be to solve this problem: Of all rectangles with a fixed perimeter, which has the greatest area?
    1. Just so that you have a number to work with, suppose your are studying the set of all rectangles that have perimeter 20 units. Let x and y be the lengths of the sides of one of the rectangles. Use the formula for the perimeter of a rectangle to find a formula for y in terms of x.
    2. Find a function A(x) that gives the area of the rectangle in terms of the side length x.
    3. Complete the following table of values.

          Side length x         Area A(x)    
      0 0 (No rectangle)
      1    
      2    
      3    
      4    
      5    
      6    
      7    
      8    
      9    
      10 0 (No rectangle)

    4. On graph paper, plot the points from the table above and draw a smooth curve connecting them.
    5. Based on your graph, what is the maximum area and what are the dimensions of the rectangle with maximum area?
    6. How would your results change if, instead of studying rectangles with perimeter 20, you study rectangles with perimeter 40? 100? P?
    7. Of all rectangles of fixed perimeter, which has the greatest area?
  2. In Problem #1, you solved a special case of our original problem-the case when the simple closed curve forms a rectangle. In solving our original problem, it suffices to consider only convex closed curves.
    1. On a clean sheet of notebook paper, draw a simple closed curve that is not convex.
    2. Since your curve is not convex, you can connect two points in the curve's interior with a line segment that lies partially outside of the curve. Do so.
    3. Mark the points where your line segment intersects your curve and sketch the mirror image of the curve about that line segment.
    4. Erase the original curve between your marked points. The new closed curve has the same length as the original, but the enclosed region has a greater area. Explain.
    5. Explain why the maximum area enclosed by a simple closed curve is necessarily done so by a convex curve.
  3. Use Wingeom to construct random convex polygons and compute their areas and perimeters.
    1. Complete the following table.

      Number of sides Perimeter Area Perimeter2 ¸ Area
      5            
      6            
      7            
      8            
      9            
      10            
      11            
      12            
      13            
      14            
      15            

    2. Based on your table, it appears that the ratio of perimeter squared to area is always greater than what number.
  4. It can be shown that if L is the length of a simple closed curve and A is the area of the region enclosed by the curve, then L2/A ³ 4p. This result is called the isoperimetric inequality. It is usually written 4 pA £ L2. Explain how the isoperimetric inequality justifies the following statement: Of all simple closed curves of fixed length, the circle encloses the greatest area.
  5. Write a brief history of the isoperimetric inequality.

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On 27 Dec 2001, 11:11.