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Maths Fencing Coursework

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Introduction

Maths Coursework

The Fencing Problem

A farmer has brought 1000 metres of fencing to make a crop area. The farmer wants the fencing to be put in a shape where it cand hold the maximum area/size. I will investigate many shapes so he can achieve the largest area.

The farmer does not mind what shape is created so firstly I am going to investigate:

Squares, rectangles & quadililaterals.

After I will investigate: Triangles, polygons and circles.

RECTANGLES

I have collected information about rectangles that show me what lenght and width is needed to get the maximum area. I have organised all my data on a spreadsheet.

To work out the area I do Length x Width.

Length: On the spreasheet I typed three numbers that went up in acsending order in 25s, I then highlighted them and dragged it down to maximum it could go to.

image59.pngimage60.png

Width: To work out the width In a column next to the length I would write for e.g. ( If I  was on the 2B slot) : = (1000/20) –A3

image68.png

Area: To work the area I have to type in the first area . Then to work ou the rest I highlight it and drag it down.        image69.pngimage70.png

image71.png

I am going work thes rectangles out to see wheather my graph is wrong or right.

...read more.

Middle

100))   =22360.67977

Side a: 250.5

Side b: 250.5

Side c: 499

√(500*(500-250.5)* (500-250.5)* (500-499))   =5578.989604

Here is a equilateral Triangle example:

image62.png

Side a: 333.333

Side b: 333.333

Side c: 333.333

√(500*(500-333.333)* (500-333.333)* (500-333.333))   =48112.285 cm ²        

TRIANGLES

I have collected information about triangles that show me what type triangle has the maximum area. I have organised all my data on a spreadsheet.

To work out the area I use Hero’s rule, Pythagoras or another formula.

Scalene

I have used Heros rule to work out the area of this Scalene triangle to ensure that the spreadsheet data gets the same results.

image61.pngimage62.png

S=500

Area= √(500*(500-3)* (500-498)* (500-499))   =704.9822693

These are the spreadsheet data results in 100’s.

a

 B

 c

s

s-a

s-b

s-c

Area

2

498

499

500

498

2

1

705.6911506

101

400

499

500

399

100

1

4466.542287

201

300

499

500

299

200

1

5468.089246

301

200

499

500

199

300

1

5463.515352

401

100

499

500

99

400

1

4449.719092

As you can see these results are not very accurate.image29.png

I now have to go in 10’s for more accurate results.

a

 B

 c

s

s-a

s-b

s-c

Area

151

350

499

500

349

150

1

5116.150897

161

340

499

500

339

160

1

5207.686627

171

330

499

500

329

170

1

5288.194399

181

320

499

500

319

180

1

5358.17133

191

310

499

500

309

190

1

5418.025471

201

300

499

500

299

200

1

5468.089246

211

290

499

500

289

210

1

5508.629594

221

280

499

500

279

220

1

5539.855594

231

270

499

500

269

230

1

5561.924127

241

260

499

500

259

240

1

5574.943946

251

250

499

500

249

250

1

5578.978401

261

240

499

500

239

260

1

5574.047004

These results show me that the maximum area for a scalene triangle may be the area of an isosceles triangle.

image30.png

I will go in 0.5’s just to see if my prediction is right.image31.png

a

 B

 c

s

s-a

s-b

s-c

Area

252

249

499

500

248

251

1

5578.888778

251.5

249.5

499

500

248.5

250.5

1

5578.944793

251

250

499

500

249

250

1

5578.978401

250.5

250.5

499

500

249.5

249.5

1

5578.989604

250

251

499

500

250

249

1

5578.978401

My prediction was right that the maximum area for a scalene triangle is an isosceles triangle.

image63.png

Isosceles          a²+b²=c²

I have used Pythagoras to work out the area of the Isosceles triangle.

Example:

image64.png

0.5² + b² = 499.5²

0.25 + b = 249500.25

249500.25 - 0.25 =249500

b² =249500

b = √249500

b/ height= 499.4997497

        =499.49 cm

Area = 499.49 x 1

                   2

Area= 249.745

...read more.

Conclusion

N= number of sides

The angle of a shape is: 360

                           2N

The length of one side is: 1000

                        2N

Height of a triangle is length of a side divideded by the angle times tan.

image49.png

1000       tan 360

 2N                2N

=     1000          

    2N tan 180

                   N

I have found what the height is so I have to times the height with the base and divide by 2.

BASE            HEIGHTimage56.pngimage57.png

1000    X       1000                    X  1I EIGHTER DIVED BY TWO OR TIMES BY 1/2

   N             2N tan 180                  2

                                  N

=        1,000,000 X N1

          2N² tan 180           X      2

                        N

=         1,000,000 X N          =    250,000  X N

            4N² tan 180                     N² tan 180

                          N                                   N

image67.png

Shape

Sides

Maximum Area

pentagon

5

68819.09602

hexagon

6

72168.78365

 heptagon

7

74161.47845

 octagon

8

75444.17382

enneagon

9

76318.81721

 decagon

10

76942.08843

 hendecagon

11

77401.9827

 dodecagon

12

77751.05849

 triskaidecagon

13

78022.2978

 tetrakaidecagon, tetradecagon

14

78237.25478

 Penta

kaidecagon, pentadecagon

15

78410.50182

 hexakaidecagon, hexadecagon

16

78552.17956

 heptakaidecagon

17

78669.52214

 octakaidecagon

18

78767.80305

enneakaidecagon

19

78850.94024

 icosagon

20

78921.89393

     From the results I have got I can see that the more sides the polygon has the higher the area, So I have decided to find out the area of a circle.

image58.png

2пr= 1000 cm                                                       r = radius

пr=1000                                                                п= 3.141592654…..

         2

r = 500

         п                          

п x (500 )  ² = 79577.47155  

          П                            

This shows me that the circle has the maximum area with a perimeter of 1000cm.

My prediction was wrong that the polygon with the most sides would have the highest area

But actually the shape with only one side (circle) has the maximum area this is something I have just learnt after doing the calculations.

To conclude I have examined many shapes such as triangles (equilateral, scalene & isosceles), quadrilaterals and many polygons. I have learnt that a one sided shape can hold the maximum area with a perimeter of 1000cm.

...read more.

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