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  • Level: GCSE
  • Subject: Maths
  • Word count: 1152

Fencing Problem

Extracts from this document...

Introduction

Simran Singh Ghatore        

11B

Mathematics

Fencing Problem

Aim-: A farmer has 100m of wire. She wishes to fence of an area of land into a shape, which will give her the maximum area.

  • I have to measure and experiment with different shapes of 1000m circumference of perimeter to achieve the maximum area.
  • I have to consider working from these shapes-:

- Quadrilaterals            

- Triangles

- Pentagon

- Hexagon

- Circle

Prediction-: I predict that if the sides of a regular shape increase so will the area, coming to the conclusion of the circle holding the most area.

Square(l x w)*NOT TO SCALE

image00.png

Rectangles (l x w)

In a rectangle, any 2 different length sides will add up to 500, because each side has an opposite with the same length. Therefore in a rectangle of 100m X 400m, there are two sides opposite each other that are 100m long and 2 sides next to them that are opposite each other that are 400m long.

image06.pngimage13.png

image22.png

Length (m)

Width (m)

Area (m2)

480

20

9600

425

50

22500

400

100

40000

350

150

52500

325

175

56875

300

200

60000

250

250

62500

...read more.

Middle

image04.png

image05.png

image08.pngimage07.png

Base (m)

Side (m)

Height (m)

Area (m2)

480

260

10000

24000

400

300

223.607

44721.4

333.33

333.33

288.675

48112

300

350

316.228

47434.2

250

375

353.553

44194.125

180

410

400

36000

150

425

418.330

31374.75

50

475

474.342

11858.55

Graph

Conclusion

The regular triangle seems to have the largest area out of all the areas but to make sure I am going to find out the area for values just around 333.

Base (m)

Side (m)

Height (m)

Area (m2)

333

333.5

 288.964

 48112.450

333.25

333.4

 288.747

 48112.518

333.3

333.4

 288.704

48112.522

333.5

333.3

 288.531

 48112.504

333.75

333.1

 288.314

 48112.410

334

333.0

 288.097

 48112.233

This has proved that once again, the regular shape has the largest area.

Because the last 2 shapes have had the largest areas when they are regular, I am going to use regular shapes from now on.

Regular Pentagon

Because there are 5 sides, I can divide it into 5 segments. Each segment is an isosceles triangle, with a top angle of 720. This is because it is a fifth of 360. This means I can work out both the other angles by subtracting 72 from 180 and dividing the answer by 2. This gives 540

...read more.

Conclusion

I am going to apply the same method as before to solve the areas of these two regular shapes.

Hexagon:

1000 ÷ 6 = 166 1/6 ÷ 2 = 83 1/3

360 ÷ 6 = 60 ÷ 2 = 30

COS= Adjacent/Hypotenuse

Adjacent= 831/3 m andHypotenuse= h

= 831/3 / h

=83.33333335/COS60

=166.6666667 = 166.66666672 – 83.33333352

= √20833.33332

h = 144.338

Area = ½ X b X H = ½ x 83 1/3 X 144.338 = 6014.065

6014.065 X 12 = 72168.784m2

image15.png

image16.png

Heptagon:image28.png

1000 ÷ 7 = 142.857 ÷ 2 = 71.429

360 ÷ 7 = 51.429 ÷ 2 = 25.714

TAN= Opposite/Adjacent

= x/71.4285 = h= 148.323

Area = ½ X b X H = ½ X 71.429 X 148.323 = 5297.260

5297.260 X 14

= 74161.644m2

Octagonimage17.png

image18.png

TAN= Opposite/Adjacent

TAN 67.5o = h/62.5

h= 62.5xTan67.5 o

=Area= ½ x 125 x h

COS= Adjancent/Hypotenuse = 62.5/x

= 62.5/COS67.5=163.320mimage19.png

TAN=62.5 x Tan67.5= 150.888m

Area= ½ x 150.888 x 125= 9430.3 x 8

=73444m2

My predictions were correct and as the number of side’s increases, the area increases. Below is a table of the number of sides against area:

No. of sides

Area (m2)

3

48112.522

4

62500.000

5

68819.096

6

72168.784

7

74161.644

8

73444

Circle

I have already come to the conclusion that: as the number of side’s increases, the area increases. Circles have no sides and are infinite. This should prove my prediction correct.

image20.png

C= 2 π x r

1000= 2 x 3.14 x r

image21.png

1000= 6.28 x r

1000  

 6.28    = r

r= 159.23

Area = πr2

= 3.14 x 159.23 x 139.23

=79612.2m2

From this I conclude that a circle has the largest area when using a similar circumference.

...read more.

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