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  • Level: GCSE
  • Subject: Maths
  • Word count: 2049

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.

...read more.

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)

...read more.

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.

image02.pngimage03.pngimage04.png

image01.pngimage05.png

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.

...read more.

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