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  • Level: GCSE
  • Subject: Maths
  • Word count: 1483

Maths:Fencing Problem

Extracts from this document...

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

image00.png

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

image01.png

...read more.

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.

image03.png

...read more.

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.

...read more.

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