# The Fencing Problem

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Introduction

Maths coursework

Maths coursework

THE FENCING PROBLEM

Investigation

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

She is not concerned about the shape of the plot, but it must have a perimeter (or circumference) of 1000m.

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

Therefore, my aim 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.

Rectangles

I will start with working out the area of rectangles as it is easy to find its area.

22,500m2 40,000m2 52,500m2 60,000m2 62,500m2

60,000m2 52,500m2 40,000m2 22,500m2

Table

Width / m | Height / m | Area / m2 |

50 | 450 | 22,500 |

100 | 400 | 40,000 |

150 | 350 | 52,500 |

200 | 300 | 60,000 |

250 | 250 | 62,500 |

300 | 200 | 60,000 |

350 | 150 | 52,500 |

400 | 100 | 40,000 |

450 | 50 | 22,500 |

Graph

The table shows that the maximum area of a rectangle with a perimeter of 1000m is a 250 x 250 square.

Proof

In the graph, if we look at one point on either side of the highest point, they are clearly less than the maximum point, which proves that a square has the highest area.

Triangles

Next, I will look at the maximum area of a triangle with a perimeter of 1000m.

I will use the cosine rule to work out one of the angles:

Cos A = (b2 + c2 – a2) / (2bc)

Cos A = (2502 + 3502 – 4002) / (2 x 250 x 350)

Middle

Cos A = 0.5 so A = 60o

Now that I have one angle, I can work out the area:

Area = ½ x b x c x Sin A

Area = ½ x 333.3 x 333.3 x Sin (60) = 48,112.5cm2

I think that the above triangle will give the highest area (for triangles) because with the rectangles, the one with the highest area was the square, which had all 4 sides the same length. Similarly, with the above triangle, all three sides were the same length and this meant that the above triangle had a higher area than the first one. However, to make sure that my prediction is right, I will work out the area of a triangle very similar to the one above.

I will use the cosine rule to work out one of the angles:

Cos A = (b2 + c2 – a2) / (2bc)

Cos A = (3402 + 326.62 – 333.32) / (2 x 340 x 326.6)

Cos A = 0.5006 so A = 59.96o

Now that I have one angle, I can work out the area:

Area = ½ x b x c x Sin A

Area = ½ x 340 x 326.6 x Sin (59.96) = 48,074.02cm2

This shows that shapes with all sides the same length have the highest formula.

Why do regular shapes have the biggest area?

All regular sided shapes have an n (n being the number of sides in that shape) number of equilateral triangles. Therefore, it is important to realise why an equilateral triangle has the biggest area compared to any other type of triangle. However,

Conclusion

Circumference = 2πr

1000 = 2πr

1000 / 2π = r

159.1549431 = r

Now I can calculate the area:

Area = πr2

Area = 79,577.4715459477cm2.

This shows that the shape with the maximum is with a perimeter (or circumference) of 1000m is a circle. I firstly worked out that as the number of sides increased, the area increased as well but at a decreasing rate. The reason that I realised that a circle would give the maximum area was because as the number of sides increased, it would at one point reach an infinite number, which would be a circle. So therefore, a circle must have the maximum area because it has the most number of sides from any other shape.

Why a circle has the maximum area

As we can see from the two pictures above, the triangle and octagon fit into the circle easily. However, there are many gaps left between the shape and the circle. This represents the extra are the circle has and answers why a circle has the maximum area.

Conclusion

Ultimately, I would recommend the farmer build a fence in the shape of a circle, with a circumference of 1000m and a maximum area of 79577.4715459477cm2. This will give her the maximum area with the length of fence that she has.

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

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