# Fencing problem.

Extracts from this document...

Introduction

Imy Jacobs Math c/w 2004

Fences Coursework

2004

A farmer has exactly 1000m of fencing. With it she wishes to fence off a plot of level land. She isn’t concerned about the shape but states that it must be 1000m in perimeter.

What she wants to know is what is the maximum area that she can have with that amount of fencing.

By drawing scale outlines, I will examine some possible shapes for the plot of lad. In each case I will need to ensure that the perimeter is 1000m, and then I need to obtain the enclosed area.

Rectangles

The first shapes that I will be investigating are rectangles. I am going to draw three rectangles and see which of the three has the bigger area.

There are different types of rectangles that I could draw, including squares.

Middle

Base/m

Sides/m

Area/m2

50

475

11 858.5

100

450

22 360.7

150

425

31 374.8

200

400

38 729.9

250

375

44 194.2

300

350

47 434.2

350

325

47 925.7

400

300

44 721.4

450

275

35 575.6

A rectangular field has a bigger area because it has more sides.

Step 3; Zooming In

So I can gain a more accurate peak in my first Triangle graph I am planning to zoom in and find the areas of some of the triangle between 300m and 350m along the base.

Three equilateral triangles

1) Area = s(s-a)(s-b)(s-c)

= 1000(1000 -1000/3)(1000 -1000/3)(1000 -1000/3)

= 48 112.52m2

2)

Conclusion

Formula; Scale;

b x h and Tan = Opposite 1cm : 100m

2 Adjacent

Polygon 1 – Pentagon

Polygon 2; Hexagon

Polygon 3; Octagon

More Polygons

No. of sides/m | Length/m | Area/m2 |

5 | 200 | 68819.1 |

6 | 1000/6 | 72168.8 |

7 | 1000/7 | 74 161 |

8 | 125 | 75444.2 |

9 | 1000/9 | 76 319 |

10 | 100 | 76 942 |

20 | 50 | 78 922 |

50 | 20 | 79 472.7 |

100 | 10 | 79 551 |

As you can see by the table the area grows larger and larger as the polygons gain more sides. This means that the more sides the greater the area so we would want to build a field with as many sides as possible.

Looking at the diagrams you may notice that as the shapes gain more sides the lengths become shorter and they look more and more like a circle. Because of this I have decided to check out a circle and see how big an area that has.

Area = πr2

C = πd

1000 = πd

d = 1000 = 318.309

π

r = 1000 = 159.15

2π

Final Conclusion

We can now see that a circle has the greatest area of all shapes that we looked at, and therefore that is the best shape for the farmer to use when she builds her field.

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

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