# Maths:Fencing Problem

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Introduction

Fencing Coursework

A farmer has 1000m of land but wants to fence off a plot of land. I am going to investigate different shapes that will give me a maximum area using 1000m of fencing.

First of all I will investigate rectangles with different lengths and widths but all add up to perimeter of 1000m. Then I will look at other shapes such as triangles and polygons. I will draw graphs and tables; after I have completed my investigation I will advise the farmer.

I will start with a rectangular shape with a base of 50m and keep increasing the pitch by 50m each time. Then I will explore other four sided shapes.

length of shape | width of shape | area=l*w |

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 |

This shows that 250 is the

Max because after 250 the

Area starts to go down

The maximum area is when the sides are 250m each. I will double check to make sure this is the real maximum, by looking at the area around the length 245 to 252. I will go up by a pitch of 1m.

Base | Height | Area |

245 | 255 | 62475 |

246 | 254 | 62484 |

247 | 253 | 62491 |

248 | 252 | 62496 |

249 | 251 | 62499 |

250 | 250 | 62500 |

251 | 249 | 62499 |

252 | 248 | 62496 |

Middle

1000-50=950 = 475m

2

475m

50m h 475

1000-100=900=450m

2

25m

450m

h

100 m

50m

I will out my calculations in a table using excel spreadsheet.

Base | 1/2 base | Square of Sides | Hyp (side) | Sq Root | area |

50 | 25 | 475 | 225000 | 474.341649 | 11858.5412 |

100 | 50 | 450 | 200000 | 447.213595 | 22360.6798 |

150 | 75 | 425 | 175000 | 418.330013 | 31374.751 |

200 | 100 | 400 | 150000 | 387.298335 | 38729.8335 |

250 | 125 | 375 | 125000 | 353.553391 | 44194.1738 |

300 | 150 | 350 | 100000 | 316.227766 | 47434.1649 |

350 | 175 | 325 | 75000 | 273.861279 | 47925.7238 |

400 | 200 | 300 | 50000 | 223.606798 | 44721.3595 |

450 | 225 | 275 | 25000 | 158.113883 | 35575.6237 |

This triangle would not be possible because the sides would not be able to join up.

This table shows that 300 is the maximum an isosceles triangle will go up to because 350 will not make a triangle. So I will try to find out if 300 base is the real maximum. I will do this by uinvestigating around the numbers 300 to 349. but firstly I will put all rhis in a graph using excel spreadsheet.

Conclusion

hypotenuse

sq root

area

330

165

335

85000

291.5475947

48105.35

331

165.5

334.5

84500

290.6888371

48109

332

166

334

84000

289.8275349

48111.37

333

166.5

333.5

83500

288.9636655

48112.45

334

167

333

83000

288.0972058

48112.23

335

167.5

332.5

82500

287.2281323

48110.71

336

168

332

82000

286.3564213

48107.88

337

168.5

331.5

81500

285.4820485

48103.73

338

169

331

81000

284.6049894

48098.24

339

169.5

330.5

80500

283.7252192

48091.42

340

170

330

80000

282.8427125

48083.26

In this table I have found a more accurate result but I am still not satisfied that this is the maximum so I will go on to investigate between 333.1 to 334. which means that I will draw another table using excel spreadsheet again. This will be my final table on isosceles triangles because after this I will know the maximum area for an isosceles triangle.

Base | 1/2 Base | sq of sides | hypotenuse | sq root | area |

333.1 | 166.55 | 333.45 | 83450 | 288.8771365 | 48112.49 |

333.2 | 166.6 | 333.4 | 83400 | 288.7905816 | 48112.51 |

333.3 | 166.65 | 333.35 | 83350 | 288.7040007 | 48112.52 |

333.4 | 166.7 | 333.3 | 83300 | 288.6173938 | 48112.52 |

333.5 | 166.75 | 333.25 | 83250 | 288.5307609 | 48112.5 |

333.6 | 166.8 | 333.2 | 83200 | 288.444102 | 48112.48 |

333.7 | 166.85 | 333.15 | 83150 | 288.3574171 | 48112.44 |

333.8 | 166.9 | 333.1 | 83100 | 288.2707061 | 48112.38 |

333.9 | 166.95 | 333.05 | 83050 | 288.183969 | 48112.31 |

334 | 167 | 333 | 83000 | 288.0972058 | 48112.23 |

This final table shows me that 333.3 is the maximum and 48112.52 is the maximum area. it seems the area keeps increasing and my isosceles triangle is beginning to look like an equilateral triangle. The red coloured boxes are the triangles which would not be possible because the base is too long so the sides’ wont join together to make a triangle as they would not be able to reach each other.

Observation:

1. I noticed that as I increased the base the other 2 sides became shorter because

My fencing is a fixed amount of 1000m.when the sides are 330 the other side Becomes longer in length.

2. Now I will move on to scalene triangles to see if they will give me a higher area. I will need to use hero’s formula.

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

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