# Fencing problem

Extracts from this document...

Introduction

Page of

Maths Coursework

## Fencing problem

I have been asked to investigate the maximum area enclosed by 1000 metres of fencing. To do this I need to use various shapes, to start with a square.

Secondly I shall look at rectangles.

Rectangle | Length (M) | Height (M) | Area (M) |

A | 450 | 50 | 22500 |

B | 400 | 100 | 40000 |

C | 350 | 150 | 52500 |

D | 300 | 200 | 60000 |

E | 250 | 250 | 62500 |

F | 200 | 300 | 60000 |

The biggest quadrilateral in the table was E which is a square, meaning the square has a larger area than rectangles. I only investigated six rectangles as after E the rectangles have the same numbers but on different sides. This means the area would be the same and so it was pointless to continue at this rate of increase so I looked

Middle

Triangle | Equal sides (m) | Base | Total area (m ) |

A | 300 | 400 | 44721.35955 |

B | 350 | 300 | 47434.1649 |

C | 400 | 200 | 38729.83346 |

D | 450 | 100 | 4330.127019 |

The biggest area I found was with triangle B although as I increased by 50m each time I think that I could investigate this triangle in more depth with smaller intervals to find a larger area.

I continued with the same formula as before but just applied it to another group of triangles. These are my results.

Triangle | Equal sides (m) | Base (m) | Total area (m ) |

A | 305 | 390 | 45731.553658 |

B | 310 | 380 | 46540.3051128808 |

C | 315 | 370 | 47165.9305007334 |

D | 320 | 360 | 47623.5235991628 |

E | 325 | 350 | 47924.7237817015 |

F | 330 | 340 | 48083.2611206854 |

G | 335 | 330 | 48105.353132 |

H | 340 | 320 | 48000 |

I | 345 | 310 | 4774.208523 |

Although I could go further into depth with Triangle G I am

Conclusion

Now that I know the perpendicular height I can use the same formula as before to get the area.

Base X height ÷ 2

125 X 150.8883476 ÷ 2 = 9430.521725

Area = 9430.521725m

This gives me the area of one of eight equal triangles in an octagon so now I must multiply this number by eight.

9430.521725 X 8 = 75444.1738

As expected the area has again increased in size. I think that for regular shapes the more sides the shape has the bigger the area is. A circle has infinite sides in theory so I think this will have the biggest area.

Radha Campbell 10L

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

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