A real estate developer is given a fence with length 1000 meters. She can fence off any piece of land she wants as long as she doesn't use more than 1000 meteres. What shape would have the largest area? - Show that if the developer puts the fence as a triangle, then among all triangles with perimeter equal to 1000, the equilateral triangle has the largest area? Hint: Look up Heron's formula and AMGM inequality. - Show that if the land is chosen to be a rectangle, then the rectangle with equal sides has the largest area. - Compare the area of a equilaterial triangle, a square and a pentagon, all with perimeter 1000. Which one has the largest area? - Consider the convex regular polygon with n sides. For n = 3 this is a equilaterial triangle, n = 4 a square and n = 5 a pentagon. Assume the perimeter of the polygon is 1000. Find the area of the polygon as a function of n. Is it increasing or decreasing? What is lim Area(n)? 71}00 - Compare the above limit to the area of a circle with perimeter 1000? which one is larger? - Among all the shapes in the world with perimeter 1000, which one has the largest area? Can you give an intuitive explanation? This is known in mathematics as the Isoperimetric Inequality. Iso=Same, Perimetric=Relating to Circumference. You can get some inspiration from the pictures on wikipedia. Isoperimetric Inequality