# The Fencing problem.

Extracts from this document...

Introduction

Plan

A farmer has exactly 1000 metres of fencing and wants to fence off a plot of level land.

The farmer is not concerned about the shape of the plot but it must have a perimeter of 1000 metres. This means that the plot could be:

Or any other shape with a perimeter (or circumference) of 1000 metres.

The farmer wishes to fence off the plot of land, which contains the maximum area.

My task is to investigate the shape, or shapes, that could be used to fence in the maximum area using exactly 1000 metres of fencing each time.

To do this experiment to a decent standard I must investigate as many different shapes as possible. From a Triangle to a Circle. I will am going to attempt to collect data from all these shapes so that I can come to a solution for the farmers problem also using all these shapes will help me come to a proof which will prove to the farmer why the shape or shapes that I have said to have the biggest area are the biggest area.

Middle

480

20

9600

490

10

4900

Rectangles

As you can seefrom the graph that it is a Parabolas graph, which means that there are quadratic equations involved. From my table and graph you can see that the highest area is a Square or a regular quadrilateral. I only measured to the nearest 10m I cannot tell if this is true. So I will test to see what the area is when the width is: 249m, 249.5m 249.75m, 250m, 250.25, 250.5, and 251m.

Width | Length | Area |

249 | 251 | 62499 |

249.5 | 250.5 | 62499.75 |

249.75 | 250.25 | 62499.94 |

250 | 250 | 62500 |

250.25 | 249.75 | 62499.94 |

250.5 | 249.5 | 62499.75 |

251 | 249 | 62499 |

Allof the results fit into the graph. This

Conclusion

Base length = P/n

½ Base length = (P/n)/2

Bottom angle of segment = ((180-(360/n))/2)

Height = ((P/n)/2)*tan((180-(360/n))/2)

Area of segment = ½ (P/n)*((P/n)/2)*tan((180-(360/n))/2)

Total area of shape = (½ (P/n)*((P/n)/2)*tan((180-(360/n))/2))

#### Number of sides to Area

Number of sides | Area |

4 | 62500 |

5 | 68819.09602 |

6 | 72168.78365 |

7 | 74161.35699 |

8 | 75444.17 |

9 | 76318.81 |

10 | 76942.09 |

15 | 78410.5 |

25 | 79158.15 |

30 | 79286.37 |

35 | 79363.64 |

45 | 79448.15 |

50 | 79472.72 |

80 | 79536.56 |

100 | 79551.29 |

200 | 79570.93 |

500 | 79576.42 |

1000 | 79577.21 |

1500 | 79577.36 |

10000 | 79577.4689 |

20000 | 79577.4708 |

40000 | 79577.4713 |

100000 | 79577.47152 |

500000 | 79577.47150000 |

The graph which shows the pattern between the increasing number of sides and the increasing area on the page before becomes an asymptote, an asymptote is a graph which gets closer and closer to a horizontal straight line but never gets to that point.

This explanation of this graph proves to me that the circle is the highest area possible because as we increase the sides the area also increases, the graph will go on forever. This is why the circle has the highest area it has an infinite number of sides. So theoretically the graph will go on forever until we reach infinity and will then stop to show the circle to be the shape with the highest area because it has infinite sides.

#### Area of a Circle

Circumference of circle = 1000 Circumference of a circle = πd

Diameter = 1000/π = 318.3098

Radius = d/2 = 159.1549

Area = πr2

Area = π*159.15492 = 79577.471553

This work has proven that the farmer should use the 1000m of fence to build a circle because it has the biggest area.

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

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