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  • Level: GCSE
  • Subject: Maths
  • Word count: 1633

The Fencing Problem

Extracts from this document...

Introduction

Adam Ziggy NiblettMaths Coursework

The Fencing Problem

Question

A farmer has exactly 1000 metres of fencing, with it she wishes to fence off a plot of level land. She is not concerned about the shape of the plot, but it must have a perimeter of 1000 metres. What she does wish to do is fence off the land which contains the maximum area.

Investigation

In my investigation I am going to calculate the area of many shapes, (Circle, Square, Rectangles, Polygons and Triangles). I am going to change the length of the sides to determine the greatest area; all the shapes will have a perimeter of 1000 metres.

Hypothesis

I predict that the circle will have the greatest are and the triangle will have the smallest area. The circle has one continual line and therefore will have the biggest area. As the amount of sides a shape has decreases the area of that shape decreases.

Squares and Rectangles

     1.                                 A=L x L            

                                     =250 x 250

                                     =62,500m

    2.                                A=L x W

                   =300 x 200

                                     =60,000m

    3.                               A=L x W

                                    =350 x 150

                                    =52,500m

    4.A=L x W

                                                  =499 x 1

                     =499m

    5.

...read more.

Middle

                                     =  2,314,953,706.48           = 48113.97m

Number of

Length of Sides

Area

Shape

(axbxc)  (m)

(m )

14

350x350x300

47,434.20

15

400x400x200

38,729.80

16

300x300x400

44,721.40

17

450x450x100

22,360.70

18

375x375x250

44,194.20

19

250x333.33x416.6

44666.3

20

333.33x333.33x333.33

48113.97 

The triangle with the largest area was the equilateral triangle. I found that as the sides became closer in length the area increased.

The square is the quadrilateral with the largest area and the equilateral triangle is the triangle with the largest area. Both these shapes have equal sides. Therefore I will only look at regular polygons, this is because they too have equal sides.  

Polygons

21. 5 sided polygon

      L= 1000                    

             5

        =200m

                            Using Tan = O

                                               A

                               Tan54  =         (x100)    

                                             100

                      100 x Tan54  =

                                  137.6 =

              Area One Triangle  = 137.6 x 200 = 27,520 = 13,760m  

                                                        2                 2

                   Area of Polygon = 13,760 x 5 = 68,800m

22. 10 sided polygon

      L= 1000                    

             10

        =100m

                            Using Tan = O

                                               A

                               Tan72  =         (x50)    

                                              50

                     50 x Tan72  =

                              153.8 =

       Area One Triangle  = 153.8 x 100 = 1,538,841.7 = 7,694.2m  

                                                 2                   2

                   Area of Polygon = 7,694.2 x 10 = 76,942.1m

23. 20 sided polygon

     L= 1000                    

           20        

        =50m

                            Using Tan = O

                                               A

                               Tan81  =         (x25)    

                                             25

                      25x Tan81  =

                                  157.8 =

              Area One Triangle  = 157.8 x 50 = 7,892.2 = 3,946m  

                                                       2                2

                   Area of Polygon = 3,946.2 x 20 = 78,921.9m

24. 50 sided polygon

     L= 1000                    

           50        

        =20m

                            Using Tan = O

                                               A

                               Tan86.4  =         (x10)    

                                                 10

                      10x Tan86.4  =

                                  158.9 =

              Area One Triangle  = 158.9 x 20 = 3,178.9 = 1,589.5m  

                                                       2                2

                   Area of Polygon = 1,589.5.2 x 50 = 79,472.7m

25. 100 sided polygon

...read more.

Conclusion

Area

Shape

(m)

(m )

21

5 sides

200

68,800 

22

10 sides

100

76,942.10

23

20 sides

50

78,921.90

24

50 sides

20

79,472.70

25

100 sides

10

79,551.30

26

200 sides

5

79,570.90

27

250 sides

4

79,573.30

28

500 sides

2

79,576.40

29

750 sides

1.3

79,577

30

1000 sides

1

79,577.20

31

2000 sides

0.5

7,957.10

32

Circle

N/A

79,577.50

Conclusion

        I have concluded that  a shape with three sides and tree angles (triangles) will have the smallest area. Shapes with four sides and four angles (equilaterals) will then be next. As the number of sides and angles increase the size will increase. This will occur until you reach the circle, which does not contain sides or angles, it has one continual line. This is why it has the largest area.

As these diagrams show, as you get closer to the shape of the circle the area increases

Isoperimetric Quotations  

Circle Area =   r                                Area x 4   =   r   x 4                                              

 Perimeter  = 2  r                                               = 4    r                            

              P   =4    r                                  4   x A = 4    r   = 1                                  

          A  P   =   r       =   1                            P        4    r                  

                     4    r       4                                                                                          

The IQ for all the shapes is defined as 4   x A

                                                                  P

For a Square

   A = 250 x 250 = 62,500m

   P = 1000m          P   = 1000 x 1000 = 1,000,000m                                                  

IQ = 4   x 62,500

          1,000,000  

      = 785,398.7  = 0.785m    

         1,000,000

For a Rectangle

  A = 300 x 200 = 60,000m

   P = 1000m          P   = 1000 x 1000 = 1,000,000m                                                  IQ = 4   x 60,000

          1,000,000  

      = 753,982.24 = 0.754m    

          1,000,000

For a Triangle

  A = 350 x 350 x 300 = 47,434.2m

   P = 1000m          P   = 1000 x 1000 = 1,000,000m                                                  IQ = 4   x 47,434.2

          1,000,000  

      = 596,075.3 = 0.5961m    

          1,000,000

For a Polygon (5 sides)

  A = 6881.9 x 5 = 34,409.6m

   P = 1000m          P   = 1000 x 1000 = 1,000,000m                                                  IQ = 4   x 34,409.6

           1,000,000  

      = 432,403.13 = 0.4324m    

          1,000,000

...read more.

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