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Fencing problem

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Introduction

Page  of

Maths Coursework

Fencing problem

I have been asked to investigate the maximum area enclosed by 1000 metres of fencing. To do this I need to use various shapes, to start with a square.

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Secondly I shall look at rectangles.

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Rectangle

Length (M)

Height (M)

Area (M)

A

450

50

22500

B

400

100

40000

C

350

150

52500

D

300

200

60000image02.png

E

250

250

62500

F

200

300

60000

The biggest quadrilateral in the table was E which is a square, meaning the square has a  larger area than rectangles. I only investigated six rectangles as after E the rectangles have the same numbers but on different sides. This means the area would be the same and so it was pointless to continue at this rate of increase so I looked

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Middle

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Triangle

Equal sides  (m)

Base

Total area (m  )

A

300

400

44721.35955

B

350

300

47434.1649

C

400

200

38729.83346

D

450

100

4330.127019

The biggest area I found was with triangle B although as I increased by 50m each time I think that I could investigate this triangle in more depth with smaller intervals to find a larger area.

I continued with the same formula as before but just applied it to another group of triangles. These are my results.

Triangle

Equal sides (m)

Base (m)

Total area (m  )

A

305

390

45731.553658

B

310

380

46540.3051128808

C

315

370

47165.9305007334

D

320

360

47623.5235991628

E

325

350

47924.7237817015image08.png

F

330

340

48083.2611206854

G

335

330

48105.353132

H

340

320

48000

I

345

310

4774.208523

Although I could go further into depth with Triangle G I am

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Conclusion

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Now that I know the perpendicular height I can use the same formula as before to get the area.

Base X height ÷ 2

125 X 150.8883476 ÷ 2 = 9430.521725

Area = 9430.521725m

This gives me the area of one of eight equal triangles in an octagon so now I must multiply this number by eight.

9430.521725 X 8 = 75444.1738

As expected the area has again increased in size. I think that for regular shapes the more sides the shape has the bigger the area is. A circle has infinite sides in theory so I think this will have the biggest area.

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Radha Campbell 10L

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