The Fencing Problem - Math Coursework :: Math Coursework Mathematics

The Fencing Problem - Math The task -------- A farmer has exactly 1000m of fencing; with it she wishes to fence off a level area of land. She is not concerned about the shape of the plot but it must have perimeter of 1000m. What she does wish to do is to fence off the plot of land which contains the maximun area. Investigate the shape/s of the plot of land that have the maximum area. Solution -------- Firstly I will look at 3 common shapes. These will be: ------------------------------------------------------ [IMAGE] A regular triangle for this task will have the following area: 1/2 b x h 1000m / 3 - 333.33 333.33 / 2 = 166.66 333.33ÂÂ ² - 166.66ÂÂ ² = 83331.11 Square root of 83331.11 = 288.67 288.67 x 166.66 = 48112.52ÂÂ ² [IMAGE]A regular square for this task will have the following area: Each side = 250m 250m x 250m = 62500mÂÂ ² [IMAGE] A regular circle with a circumference of 1000m would give an area of: Pi x 2 x r = circumference Pi x 2 = circumference / r Circumference / (Pi x 2) = r Area = Pi x rÂÂ ² Area = Pi x (Circumference / (Pi x 2)) ÂÂ ² Pi x (1000m / (pi x 2)) ÂÂ ² = 79577.45mÂÂ ² I predict that for regular shapes the more sides the shape has the higher the area is. A circle has infinite sides in theory so I will expect this to be of the highest area. The above only tells us about regular shapes I still haven't worked out what the ideal shape is. Width (m) Length (m) Perimeter (m) Area (mÂÂ ²) 500 0 1000 0 490 10 1000 4900 480 20 1000 9600 470 30