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

# The Fencing Problem

Extracts from this document...

Introduction

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.

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

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

### 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

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