## Gcse Maths Coursework Fencing

## Math Coursework - The Fencing Problem

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The Fencing Problem Aim - to investigate which geometrical enclosed shape would give the largest area when given a set perimeter. In the following shapes I will use a perimeter of 1000m. I will start with the simplest polygon, a triangle. Since in a triangle there are 3 variables i.e. three sides which can be different. There is no way in linking all three together, by this I mean if one side is 200m then the other sides can be a range of things. I am going to fix a base and then draw numerous triangles off this base. I can tell that all the triangles will have the same perimeter because using a setsquare and two points can draw the same shape. If the setsquare had to touch these two points and a point was drawn at the 90 angle then a circle would be its locus. Since the size of the set square never changes the perimeter must remain the same. [IMAGE] The area of a triangle depends on two things: the height and the base. The base is fixed in this example so the triangle that has the biggest height, i.e. the middle triangle, will have the biggest area. The middle triangle turns out to be an icosoles triangle. I am going to focus only on icosoles triangles. I have constructed a formula linking all three sides in and icosoles triangle. [IMAGE] X X X=any number which is greater than 250 and less than 500 ======================================================== 1000 - 2X Using Pythagoras theorem I can find and equation linking a side to the area. ====================================================================== Â½(1000 - 2X)Â² + HÂ² = XÂ² HÂ² = XÂ² + (X -500)Â² H = height X 500 - X XÂ² - (500-X)Â² H Area 251 249 1000 31.6 7874.1 300 200 50000 223.61 44721.0 333.33 ## How to Cite this Page
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Math Coursework - The Fencing Problem Essay - The Fencing Problem Introduction A farmer has exactly 1000 metres of fencing and wants to use it to fence a plot of level land. The farmer was not interested in any specific shape of fencing but demanded that the understated two criteria must be met: · The perimeter remains fixed at 1000 metres · It must fence the maximum area of land Different shapes of fence with the same perimeter can cover different areas. The difficulty is finding out which shape would cover the maximum area of land using the fencing with a fixed perimeter.... [tags: Math Coursework Mathematics] | 657 words (1.9 pages) | Strong Essays | [preview] | ||

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Math Coursework - The Fencing Problem Essay - The Fencing Problem Aim - to investigate which geometrical enclosed shape would give the largest area when given a set perimeter. In the following shapes I will use a perimeter of 1000m. I will start with the simplest polygon, a triangle. Since in a triangle there are 3 variables i.e. three sides which can be different. There is no way in linking all three together, by this I mean if one side is 200m then the other sides can be a range of things. I am going to fix a base and then draw numerous triangles off this base.... [tags: Math Coursework Mathematics] | 1214 words (3.5 pages) | Strong Essays | [preview] | ||

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The Fencing Problem Essay - The Fencing Problem Introduction I am going to investigate different a range of different sized shapes made out of exactly 1000 meters of fencing. I am investigating these to see which one has the biggest area so a Farmer can fence her plot of land. The farmer isnÂ’t concerned about the shape of the plot, but it must have a perimeter of 1000 meters, however she wishes to fence off the plot of land in the shape with the maximum area. Rectangles I am going to look at different size rectangles to find which one has the biggest area.... [tags: Papers] | 2291 words (6.5 pages) | Strong Essays | [preview] | ||

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Fencing Problem Essay - Fencing Problem Introduction: A farmer has 1000 metres of fencing, and he wants to be able to get the maximum amount of space that he can with the 1000 metres. By using formulae that we already know, we can find out what shape can give the most area. I will test each shape to find the maximum area I can, then I can eventually use the shape that can create the largest area with 1000 metres of fencing. First I will look at Rectangles, then all 3 types of triangles, Pentagons, hexagons and finally circles, and by the end I will have found out which is the suitable shape to have the fence in.... [tags: Papers] | 1176 words (3.4 pages) | Strong Essays | [preview] |

### Related Searches

Math Coursework Fencing Triangle Base Theorem Shapes Sides

166.67

83330

288.97

48112.5

350

150

100000

316.23

47434.0

400

100

150000

387.30

38729.8

450

50

205000

452.77

22638.5

499

1

249000

499.00

499.00

As you can see from the table the maximum area is when X = 333 1/3.

When this number is plugged into the formula we see that this is

actually an equilateral triangle.

The next simplest shape is the 4-sided shape, namely a rectangle. I

have constructed another formula linking the sides.

500 - X

[IMAGE]

X

X

500 - X

From the diagram the area must equal -XÂ² +500X

Unlike in the previous example, this turns out to be a quadratic

equation so I can plot it on a graph.

As you can see from the graph the maximum point is when X = 250. When

this number is plugged into the formula the rectangle is really a

square.

What do a square and an equilateral triangle have in common? They are

both regular shapes i.e. all angles equal, all sides equal.

Why is this?

Make sides same length

Make sides same length

[IMAGE]Lets take the triangle example first. When you make one side

longer you will make the other shorter. This will decrease the height,

which means the area will be smaller. When both sides are the same

length they extend the height to its highest possible. Why does an

equilateral triangle have a larger area than an icosoles triangle? you

could think of it like this.

[IMAGE]

Rotate triangle onto side

Equilateral triangle

Lets take the square example. Obviously the longer the sides the

bigger the area. This means the bigger the length and the bigger the

height, the bigger the area. In this investigation we have been given

a set perimeter. To make the length longer means you have to sacrifice

the height. To make the height bigger you have to sacrifice the

length. To get the biggest area you need the sides to be as long as

possible. When the sides are equal, it means that the sides are at the

biggest they could be simultaneously. This means The closer the sides

are in the ratio of 1:1, the bigger the area. Shapes with a ratio of

sides that is 1:1 are said to be regular. Regular shapes have numerous

properties; they can be split up into icosoles triangles. Irregular

polygons can be only split up into scalene triangles. I have already

proved why icosoles triangles have a larger area than scalene. This

means regular shapes will have a larger area than irregular shapes.

From now on I am going to find out which shape has the biggest area

with a given perimeter. I will investigate only the regular shapes

because I have proved that the regular polygon has the biggest area

out of all the irregular polygons with the same perimeter.

[IMAGE]Triangle

===============

333.3

Text Box: 333.3

333.3

Using trigonometry. Area = Â½ x 333.3 x 333.3 x sin 60 = 48112.5

[IMAGE]Square

=============

250

250

250

250

Area = 250 x 250 = 62500

Pentagon

========

[IMAGE]

Each side = 200

Using trigonometry. 200/sin 72 = Y /sin 54

Y=170.1

Area = 5x( Â½ x 170.1 x 170.1 x sin 72)

= 68794.7

Hexagon

=======

[IMAGE]

Each side = 166 2/3

Using trigonometry

Â½ x 6 x 166 2/3 x 166 2/3 x sin 60

Area = 72168.8

Septagon

========

Each side = 142.9

142.9/sin 51.4 = Y/sin 64.3

Y = 164.8

Area = 7x( Â½ x 164.8 x 164.8 x sin 51.4)

Area = 74288.7

Octagon

=======

Each side = 125

125/sin 45 = Y/sin 67.5

Y = 163.3

Area = 8x( Â½ x 163.3 x 163.3 x sin 45)

Area = 75425.4

Lets put all these results in a table

Number of sides

Maximum area with perimeter of 1000M

3

48112.5

4

62500.0

5

68794.7

6

72168.8

7

74288.7

8

75425.4

As you can see these results will keep on increasing and increasing.

This means the shape that can have the largest area must have infinite

sides. What shape has infinite sides? I will use the regular polygon

symmetry theorem.

A triangle has three sides and it has 3 lines of symmetry.

A square has 4 sides and it has 4 lines of symmetry.

A pentagon has 5 sides and 5 lines of symmetry.

A hexagon has 6 sidesâ€¦.

You get the idea. The shape with infinite sides must have infinite

lines of symmetry. The only shape that has infinite lines of symmetry

is the circle. Lets find out the area of a circle with circumference

1000.

2pr = 1000

pr = 500

r = 159.2

A=prÂ²

A = p159.2Â²

A = 79622.53

Lets add this to our table of results.

Number of sides

Maximum area with perimeter of 1000M

3

48112.5

4

62500.0

5

68794.7

6

72168.8

7

74288.7

8

75425.4

Â¥

79622.5

The circle has the biggest area with a 1000M perimeter out of all the

polygons.

Why is this?

When a shape is split up into triangles, the more sides it has, the

more triangles there will be yet these triangles will become smaller

as the number of sides increase. The amount at which the area of the

triangle decreases is not as great as the amount the side increases

by. When you split the shape into triangle, the more sides the shape

has the smaller the angle gets in between the two equal sides but the

perimeter of these triangles increase as the shape has more sides. The

higher the perimeter, the larger area you can make providing the

perimeter is well used i.e. the triangle is in the form of an icosoles

triangle. A circle would have infinite sides and its angles are bigger

since bigger angles can encompass more.

Hi Louise,

You didn't mention calculus so I am going to solve this for you without using calculus. I am going to assume that the field is a rectangle and has sides of length x meters and y meters.

Since the farmer has 1000 m of fencing, 2x + 2y = 1000 and thus x + y = 500.

If x and y are equal they are each 250 m and the field is a square. I want to think in terms of "How close is the field to a square?" Suppose that x = 250 + h then, since x + y = 500, y = 250 - h. Hence the area is

(250 + h) (250 - h) = 250

^{2}- h^{2}square meters

In this form you can plainly see that 250^{2} - h^{2} is a maximum when h = 0, that is when the field is a square.

Penny

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