# The Fencing problem.

Extracts from this document...

Introduction

MATHS COURSEWORK:

## THE FENCING PROBLEM

We are told that a farmer has exactly 1000 metres of fencing with which she wishes to fence off a plot of land. The shape of the fence does not concern her; however, what she does wish to do is fence off the plot of land, which contains the maximum area. Therefore, the point of this investigation is to investigate the shape, or shapes of the plot of land that have the maximum area.

Starting with rectangles, any 2 different length sides will add up to 500, because each side has an opposite with the same length. Therefore, in a rectangle of 100m X 400m, there are two sides opposite each other that are 100m long and two sides next to them that are opposite each other that are 400m long. This means that you can work out the area if you only have the length of one side. Here are four types of squares or rectangles that all have the same perimeter but a difference in area. NOT DRAWN TO SCALE:

To work out the area of a rectangle with a width length of 150m I would subtract 150m from 500m, which would then leave me with 350m, and then I would multiply 150m by 350m giving me a total area of 52 500m2.

Middle

445.0

441.588

24287.342

120

440.0

435.890

26153.394

130

435.0

430.116

27957.557

140

430.0

424.264

29698.485

150

425.0

418.330

31374.751

160

420.0

412.311

32984.845

170

415.0

406.202

34527.163

180

410.0

400.000

36000.000

190

405.0

393.700

37401.537

200

400.0

387.298

38729.833

210

395.0

380.789

39982.809

220

390.0

374.166

41158.231

230

385.0

367.423

42253.698

240

380.0

360.555

43266.615

250

375.0

353.553

44194.174

260

370.0

346.410

45033.321

270

365.0

339.116

45780.727

280

360.0

331.662

46432.747

290

355.0

324.037

46985.370

300

350.0

316.228

47434.165

310

345.0

308.221

47774.209

320

340.0

300.000

48000.000

330

335.0

291.548

48105.353

340

330.0

282.843

48083.261

350

325.0

273.861

47925.724

360

320.0

264.575

47623.524

370

315.0

254.951

47165.931

380

310.0

244.949

46540.305

390

305.0

234.521

45731.554

400

300.0

223.607

44721.360

410

295.0

212.132

43487.067

420

290.0

200.000

42000.000

430

285.0

187.083

40222.817

440

280.0

173.205

38105.118

450

275.0

158.114

35575.624

460

270.0

141.421

32526.912

470

265.0

122.474

28781.504

480

260.0

100.000

24000.000

490

255.0

70.711

17324.116

500

250.0

0.000

0.000

Base (m)

Side (m)

Height (m)

Area (m2)

333

333.5

288.964

48112.450

333.25

333.4

288.747

48112.518

333.3

333.4

288.704

48112.522

333.5

333.3

288.531

48112.504

333.75

333.1

288.314

48112.410

334

333.0

288.097

48112.233

Base (m) | Side (m) |

Conclusion

Now that I have proved my formula works, I can use it for investigating shapes with many more sides that would have been hard to work out by drawings. Here is a table of shapes areas using the formula.

No. of sides | Area (m2) |

9 | 76318.817 |

10 | 76942.088 |

20 | 78921.894 |

50 | 79472.724 |

100 | 79551.290 |

200 | 79570.926 |

500 | 79576.424 |

1000 | 79577.210 |

2000 | 79577.406 |

5000 | 79577.461 |

10,000 | 79,577.46893 |

100,000 | 79,577.47152 |

1,000,000 | 79,577.47155 |

10,000,000 | 79,577.47155 |

The area of a circle = π x r²

The circumference of a circle = π x D

Therefore if we know the circumference of the circle (1000m), and we know π, then we can find D.

D = circumference/ π

= 1000/ π

Now that we know the diameter, we can find the radius by halving it. We do this because we need to find ‘r’ so that we can find the area of the circle.

Therefore: r = ½ x 1000/ π

Area of circle = π x ((1/2 x 1000/ π)²)

= 79,577.47155m² (to 5 d.p)

This answer is the same as my answers to all the sides above 1,000,000 sides. The reason for them being the same is because of the calculators rounding up. Therefore the only way that I can prove that the circle has the biggest area is because of its theory of having an infinite amount of sides, and I have proved that the more sides there are the bigger the area. This diagram shows that as the number of sides increase the length of them decrease, which make the area within the circle decrease.

Therefore, I conclude that the farmer should choose a circle fence of 1000m to fence of the plot of land, as it is the maximum amount of possible 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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