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  • Level: GCSE
  • Subject: Maths
  • Word count: 2540

Math Coursework Fencing

Extracts from this document...

Introduction

image00.png


image39.png

The Fencing Problem

Introduction

The fencing problem involves finding a solution for a farmer who wants to fence a levelled piece of land. The farmer has exactly 1000m of fencing therefore he wants to achieve the maximum possible area. This will require me to test all shapes and see which shape gives the biggest area. The main shapes that I will look at are the following;

  1. Rectangles
  2. Trapeziums
  3. Parallelograms
  4. Kites
  5. Triangles
  6. Polygons greater than 4 sides
  7. Circles

Rectangles

In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles.

From this definition, it follows that a rectangle has two pairs of parallel sides of equal length; that is, a rectangle is a parallelogram. A square is a special kind of rectangle where all four sides have equal length; that is, a square is both a rectangle and a rhombus.

The area of a rectangle is the product of its length and its width; in symbols,image40.pngimage40.png. For example, the area of a rectangle with a length of 5 and a width of 4 would be 20, because 5 × 4 = 20.
I first started by drawing two rectangles and finding out their area using the formula:

image01.png

A table will be best to record my results

Length=L (m)  

Width=500-L (m)

A=LxW (m)2

0

500

N/A

50

450

22500

100

400

40000

150

350

52500

200

300

60000

250

250

62500

300

200

60000

350

150

52500

400

100

40000

450

50

22500

500

0

N/A

249.99

250.01

62499.9999

250.99

249.01

62499.0199

...read more.

Middle


image52.pngimage52.png=h

4image53.pngimage53.png

Apply parallelogram formula for area

image54.png
image55.png
image56.png

It is clear that a table I necessary as it will allow me to work out the best parallelogram in the quickest manner. Because I am dealing with angles here, I will need to create many tables to test out all possible combinations of lengths, widths and angles.

Angle

30°

60°

90°

Length m

450

450

450

Width m

50

50

50

Height m

25

43.3

50

Area m²

11250

19485

22500

Angle

30°

60°

90°

Length m

400

400

400

Width m

100

100

100

Height m

50

86.6

100

Area m²

20000

34640

40000

Angle

30°

60°

90°

Length m

350

350

350

Width m

150

150

150

Height m

75

129.9

150

Area m²

26250

45465

52500

Angle

30°

60°

90°

Length m

300

300

300

Width m

200

200

200

Height m

100

173.2

200

Area m²

30000

51960

60000

Angle

30°

60°

90°

Length m

250

250

250

Width m

250

250

250

Height m

125

216

250

Area m²

31250

54125

62500

The highlighted cells again show the shape that gives the highest. This shape is a square thus again proving my prediction that the square gives the biggest area so far.


Shown Geometricallyimage15.png


image16.png

Transition

image17.png

Giving biggest area

Kites

In geometry, a kite, or deltoid, is a quadrilateral with two pairs of equal adjacent sides. Technically, the pairs of sides are disjointcongruent and adjacent. This is in contrast to a parallelogram, where the equal sides are opposite.

I predict that the closer the shape gets to a square, the bigger area it will provide.

image57.png

The general formula for calculating the area for a kite is:

  • If d1 and d2 are the lengths of the diagonals, then the area is

image58.png

...read more.

Conclusion

The area of a circle is calculated using this formula:

image107.png

The circumference of a circle is calculated using this formula:

image108.png

Since there is only 1 circumference possible we will only be able to get one area.

We don’t know the radius but because the perimeter,

image109.png

The formula can be rearranged to give us the radius:

image109.png
image110.png
image111.png

Plug r into area of a circle formula:

image112.png

Simplify:

image113.png

image115.png


image116.png

image117.png

The area of a circle with a circumference of a 1000m is approximately, 79577.47155m². So far the circle gave us the largest area.

This is because the circle has infinite number of sides and comparing the two formulas shows that if n was to be infinity it would have the biggest area.

image118.pngimage118.pngimage119.pngimage119.png

Proving Geometrically

image30.pngimage31.png

Transaction

image32.png

Giving us a bigger area

The proof above shows that as number of sides tends to infinity the area of the polygon will tend to that of a circle’s. Therefore, when n is an infinite number of sides, its area will be equivalent to that of a circles and this the largest possible area for the given perimeter of a 1000m.

In conclusion; if the farmer wanted the shape with the maximum area and a perimeter of 1000m, he ought to fence his land in the shape of a circle.

image33.png | The Fencing Problem

...read more.

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