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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

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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