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Fencing problem.

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Introduction

Number Coursework

Problem:  Farmer Pickles has 1000 metres of fencing. Find the field with the largest area encompassed by the 1000 metre fence for Farmer Pickles.

Rectangles

I started by looking at rectangles and these are the ones I looked at:

50 length – 450 height

100 length – 400 height

150 length – 350 height

200 length – 300 height

250 length – 250 height

Once I had worked out the areas of all of those rectangles I drew a graph to show my results:

image00.png

I noticed that the area became larger as the shape became more like a regular quadrilateral. With a 1000 metres of fencing you could create a quadrilateral field with a maximum of 62500m².

Triangles

I decided to look at triangles next. Right angle triangles

...read more.

Middle

  1. = 200

   5

                                        From here I used trigonometry:

                                         Tan θ = O                                                                                                 A

                                        Tan 36 = 100

                                                         A

                                        A = 100

                                          0.7265425

                                                   A = 137.6382

Then I did:

Area = (½ x 200 x 137.6382) x 5                  

Total area of regular pentagon = 68819.1

By now I saw a pattern emerging. The more sides a shape had the larger the area, even though the circumference remained 1000m. One problem that emerged was that I couldn’t keep working out things long hand so I had to develop a formula so that I could work out the area of a shape with any number of sides.

Formula

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Conclusion

               n² Tan 180

                           n

Area = 250,000

            n Tan 180

                      n

By using the final simplified formula you can work out the area to any 2D shape, and here is a table to prove it:

Sides of regular shape

Area in m²

3

4

5

10

100      

999

48,112.33

62,500

68,819.1

76946.753

79567

79574.92

As the number of sides increase so does the area, as the table proves, but also the shapes are becoming more circular. Also the formula doesn’t work with circles so I have decided to check them out.

C = 2πr

1000 = 2πr

1000 =  r

  2π

r = 159.154 m

Conclusion

As the table proves, no 2D shape has more area than a circle while still having a 1000m circumference. For Farmer Pickles a circular field would be the best option because he wanted the field with the maximum area. Also a circular fence would be relatively easier to create than a 100 sided fence.  

...read more.

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