Math Coursework - The Fencing Problem :: Math Coursework Mathematics

The Fencing Problem Introduction A farmer has exactly 1000 metres of fencing and wants to use it to fence a plot of level land. The farmer was not interested in any specific shape of fencing but demanded that the understated two criteria must be met:  · The perimeter remains fixed at 1000 metres  · It must fence the maximum area of land Different shapes of fence with the same perimeter can cover different areas. The difficulty is finding out which shape would cover the maximum area of land using the fencing with a fixed perimeter. Aim The aim of the investigation is to find out which shape or shapes of fencing will cover the maximum area of land using exactly 1000 metres of fencing material. Prediction I am predicting that the maximum area of land covered will be achieved by using the fencing shapes with the greatest number of sides. Method I made a list of possible different shapes to be investigated and assigned measurements to the sides of the shapes making sure that they fit in within the perimeter of 1000 metres of fencing. I then worked out the areas of each shape using known mathematical formulae and techniques such as Pythagoras' theorem to calculate the sides of right angled triangles; using trigonometrical functions (sine, tangent and cosine) to calculate either angles or sides of triangles constructed. Sometimes there are no known exact formulae for working out the area of certain shapes such as octagon and more complex polygons. In such cases, given shapes are split into shapes that have known formulae for areas and the worked out the areas are added together. Areas of the following shapes were investigated: square, rectangle, kite, parallelogram, equilateral triangle, scalene triangle, isosceles triangle, right-angled triangle, rhombus, pentagon, hexagon, heptagon and octagon. Results The results of the analysis are shown in Table 1 and Fig 1. Table 1 showing the areas for the different shapes formed by using the