# A farmer has 1000 metres of fencing. She wants to use it to fence off a field. The fencing has to enclose the maximum area possible; my task is to find the shape and dimension that will give the largest area

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Introduction

Introduction A farmer has 1000 metres of fencing. She wants to use it to fence off a field. The fencing has to enclose the maximum area possible; my task is to find the shape and dimension that will give the largest area. I can only do this by trail and improvement; I will use a logical sequence to help eliminate the infinite amount of shapes starting from the lowest amount of sides to the highest, and fixing values. The first shape I will investigate will be triangles. Triangle Because there is an infinite amount of triangles like scalene, isosceles, equilaterals and right-angled, I will fix the base variable so I will be able to identify any patterns from the result. The formula I will use to calculate the area of each triangle is heroes' formula S is all the sides of the triangle divided by two which is the same as the perimeter divided by two. For the next set of results I will fix the base at 400 meters. Answer = 0cm2 Answer = 29,580.39892cm2 Answer = 38,729.83346cm2 Answer = 43,301.27019cm2 Answer = 44,721.35955cm2 Answer = 43,310.27019cm2 Answer =38,729.83346cm2 Answer =29,580.39892cm2 Answer = 0cm2 Table of results: Per 1000 A B Base 100 500 400 0 150 450 400 29580.4 200 400 400 38729.83 250 350 400 43301.27 300 300 400 44721.36 350 250 400 43301.27 ...read more.

Middle

There will be no point for me to work with parallelograms that does not have right-angled triangles because the perpendicular height will not be at its upper bound. I will prove this theory again for conformation. Answer= 40000m2 The parallelograms I have just tested both have the same side length but different areas because the perpendicular height has changed. The rectangle has the upper bound of perpendicular height because the sides' area all right angled to each other. Now I will continue my investigation with different rectangles. I will use the formula Base x Height 0 x 500 = 0m2 50 x 450 = 22500m2 100 x 400 = 40000m2 150 x 350 = 52500m2 200 x 300 = 60000m2 250 x 250 = 62500m2 300 x 200 = 60000m2 350 x 150 = 52500m2 400 x 100 = 40000m2 450 x 50 =22500m2 500 x 0 = 0m2 Table Per 1000 0 500 0 50 450 22500 100 400 40000 150 350 52500 200 300 60000 250 250 62500 300 200 60000 350 150 52500 400 100 40000 450 50 22500 500 0 0 Largest area 62500 Graph Analyse I analyse that the biggest shape happened to be a square. This makes sense because both the equilateral triangle and square has the largest area and both are regular shapes however the square has more area then the triangle. ...read more.

Conclusion

part of the formula. 180/1 becomes 180 and if use with tan will become 0, this can be proven by the tan graph, but I will not go into how tan works in this topic of work. Working The perimeter of the circle is 1000m and we want to know the radius of the circle to work out the area. To work out the area of a circle you use the formula 2?r. 2 ? r = 1000 Now I will rearrange the formula. r = 1000 2 ? Now I will simplify it. r = 500 ? = 159.1549431m Now I use ?r2 to work out the area of the shape 159.1549431 x 159.1549431 x ? = 79,577.47155m2 Analyse The circle is the biggest shape of all the shapes I have tested and I think it is the absolute largest shape, but it contradicts one of my theories "The more sides the shape has the larger the area." therefore I have proven which of my statements are correct and which on is incorrect. Conclusion After my extensive research the farmer should make a circular shape with his fence to make the largest area, however you can not make a perfect circle with fencing so close to circle with do little difference to the area of the shape as shown in my results. ?? ?? ?? ?? Ahmed Agabani 10N ...read more.

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