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Find out the greatest area that can be enclosed with 1000m of fence

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Introduction

Fence

The exercise is to find out the greatest area that can be enclosed with 1000m of fence. My first impression is that the circle has the largest area, but this needs to be investigated.

Starting with rectangles, any 2 different length sides will add up to 500, because each side has an opposite with the same length. So in a rectangle of 100m X 400m, there are two sides opposite each other that are 100m long and 2 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. The equation to work out a rectangle is

1000 = width(500 –width)

Using 10m increments here is a table of areas For a graph of Height/Area  see graph 1.

Height (m)

Width

Area (m2)

0

500

0

10

490

4900

20

480

9600

30

470

14100

40

460

18400

50

450

22500

60

440

26400

70

430

30100

80

420

33600

90

410

36900

100

400

40000

110

390

42900

120

380

45600

130

370

48100

140

360

50400

150

350

52500

160

340

54400

170

330

56100

180

320

57600

190

310

58900

200

300

60000

210

290

60900

220

280

61600

230

270

62100

240

260

62400

250

250

62500

260

240

62400

270

230

62100

280

220

61600

290

210

60900

300

200

60000

310

190

58900

320

180

57600

330

170

56100

340

160

54400

350

150

52500

360

140

50400

370

130

48100

380

120

45600

390

110

42900

400

100

40000

410

90

36900

420

80

33600

430

70

30100

440

60

26400

450

50

22500

460

40

18400

470

30

14100

480

20

9600

490

10

4900

500

0

0

This shape is a parabola. If you look at the table and the graph, the rectangle with a height of 250m has the biggest area.

...read more.

Middle

This is also a parabola, therefore a square is the rectangle with a larger area, I can try this with other regular polygons

          I will try isosceles triangles because I can vary the base easily with this, I believe that the biggest area will belong to the equilateral triangle

The formula for working the isosceles triangle out is simple. Split the base in half giving two right angles. Use Pythagoras to work out the triangles height using the base length divided by two and the other side length.

...read more.

Conclusion

No. of sides

Area (m2)

20

78921.894

50

79472.724

100

79551.290

200

79570.926

500

79576.424

1000

79577.210

2000

79577.406

5000

79577.461

10000

79577.469

20000

79577.471

50000

79577.471

100000

79577.471

         From the first graph, the line straightens out as the number of side’s increases. Because I am increasing the number of sides by large amounts and they are not changing much, this indicates a graph of this would continue this trend.

 I am going to see what the result is for a circle. Circles have an infinite amount of sides, so I cannot find the area by using the equation that I have used to find the other amount of sides out. I can find out the area of a circle by using pi.

  • 1000/pi = diameter
  • diameter / 2 = radius
  • pi X radius squared = area
  • which for a 1000m circumference  79577.47155 metres squared

This is larger than any number of sides and would fit on the graph, but this difference can become almost imperceptible.

Therefore, I have proved and conclude that the polygon which will give the largest area, for a perimeter of 1000 metres is the circle. It is also quite safe to conclude that this is the same for any perimeter because the size of perimeter and scale in theory make no difference.

...read more.

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