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  • Level: GCSE
  • Subject: Maths
  • Word count: 3635

The Fencing Problem

Extracts from this document...

Introduction

The Fencing Problem

A farmer has exactly 1000 metres of fencing and wants to fence off a plot of level land.

She is not concerned about the shape of the plot, but it must have a perimeter (or circumference) of 1000m.

She wishes to fence off the plot of land, which contains the maximum area.

I am going to investigate the shape or, shapes, that have the maximum area with a perimeter of a 1000 metres.  I am going to first investigate triangles, because they are polygons with the least number of sides. Then I will progress to other polygons with increasing number of sides.  Finally, I will work out the area of a polygon with infinite number of sides, i.e. a circle, with the circumference of 1000m.


Triangles

I will investigate isosceles triangles because triangles such as scalene have more than one different variable so there are lots of possible combinations. If I know the length of the base of an isosceles triangle, I can work out the lengths of the other two sides because they are the equal.  For example, if the base is 200m, the sides would be 400m long.  To work this out I used the following formula:

length of side = (1000-base )/2

This is a diagram showing the lengths of a triangle.

image18.pngimage00.pngimage01.png

b = base of triangle

...read more.

Middle

333.4

333.30

288.617

48112.520

333.5

333.25

288.531

48112.504

333.6

333.20

288.444

48112.476

333.7

333.15

288.357

48112.435

333.8

333.10

288.271

48112.381

333.9

333.05

288.184

48112.314

334.0

333.00

288.097

48112.233

Using this results table I can draw a final graph of area against base of triangle. From the graph, I can see that the maximum area of a triangle with a range of bases, ranging from 333m to 334m, is a triangle with a base of 333.3m, with sides of 333.3m long and a perimeter of 1000m.The area of this triangle is 48112.522m².  The triangle is an equilateral triangle.


Quadrilaterals

I will investigate if and how the interior angle of a parallelogram effects the area of the parallelogram and see if the parallelogram has a bigger area than a rectangle with the same perimeter of 1000m.  To work out the area of a parallelogram, I will use the following formula:

l = length of parallelogram

w = width of parallelogram

h = perpendicular height of parallelogram

a = area of parallelogramimage14.png

image15.pngimage10.pngimage16.pngimage02.png

image11.pngimage13.png

a = l * h

However, because I only know the length and width of the parallelograms, I will need to work out the perpendicular height.  For example, if the length of a parallelogram was 300m and the width of a parallelogram was 200m, I wouldn't be able to use the formula shown above. The formula to work out the perpendicular height is:

image14.png

image10.png

image15.pngimage17.pngimage16.png

image13.png

image11.png

image20.pngimage19.png

h = w * sinx°

I used sine because I know the hypotenuse (width) and I want to work out the opposite (perpendicular height). I will now work out the area of the following parallelogram.


image21.png

image22.png

image23.png

h         = 200m * sin45°

= 141.4mimage19.pngimage15.pngimage13.pngimage16.pngimage10.pngimage17.png

a        = 300m * 141.4m

= 42420m²

I will change the angle x but keep the length and width the same to find the interior angle needed to create the maximum area.  The results were as followed:

height/m

area/m²

10

34.73

10418.89

20

68.40

20521.21

30

100.00

30000.00

40

128.56

38567.26

50

153.21

45962.67

60

173.21

51961.52

70

187.94

56381.56

80

196.96

59088.47

90

200.00

60000.00

100

196.96

59088.47

110

187.94

56381.56

120

173.21

51961.52

130

153.21

45962.67

140

128.56

38567.26

150

100.00

30000.00

160

68.40

20521.21

170

34.73

10418.89

These results show that a parallelogram has the greatest area when angle x is 90° i.e. when it is a rectangle.  Therefore, a rectangle has a greater area than a parallelogram.

I will investigate the areas of different rectangles, that each have the same perimeter of 1000 metres, because a rectangle is an easier shape for which to work out the area for than any other quadrilaterals.  To work out the area, I will use the following formula:


l = length of rectangle

w = width of rectangle

a = area of rectangle image25.pngimage27.pngimage26.pngimage29.pngimage28.png

a = l * w

The two examples I worked out are shown below.

image31.pngimage30.png

image37.pngimage36.pngimage35.pngimage32.pngimage34.pngimage33.png

image38.png

The area of the rectangles above are as follows.

  1. area = length * width

area = 150m * 350m

area = 525000m²

  1. area = length * width

area = 200m * 300m

area = 60000m²

image39.png

The examples above show that not all rectangles with the same perimeter have the same area.

The formula to work out the area of any rectangle, with a perimeter of 1000m for any given length is:

width        = 500 - length

area        = length * width

Substituting the width in the formulas above creates this formula:

area        = length * (500 - length)

area        = (500 * length) - length²

I'm going to use the formula shown above to find the maximum area of a rectangle with a perimeter of 1000 metres.  To save time and go through all the possible measurements by a calculator, I'm going to make a spreadsheet.  I will change the length of the rectangle in steps of 10m.  The formulas I will use in the spreadsheet are:

width        = 500 - length

area        = length * width

The results are as followed:

length/m

width/m

area/m²

0

500

0

10

490

4900

20

480

9600

30

470

14100

40

460

18400

50

450

22500

60

440

26400

70

430

30100

80

420

33600

90

410

36900

100

400

40000

110

390

42900

120

380

45600

130

370

48100

140

360

50400

150

350

52500

160

340

54400

170

330

56100

180

320

57600

190

310

58900

200

300

60000

210

290

60900

220

280

61600

230

270

62100

240

260

62400

250

250

62500

260

240

62400

270

230

62100

280

220

61600

290

210

60900

300

200

60000

310

190

58900

320

180

57600

330

170

56100

340

160

54400

350

150

52500

360

140

50400

370

130

48100

380

120

45600

390

110

42900

400

100

40000

410

90

36900

420

80

33600

430

70

30100

440

60

26400

450

50

22500

460

40

18400

470

30

14100

480

20

9600

490

10

4900

500

0

0

...read more.

Conclusion

a = n * ((500/n) * ((500/n) * tan(90-(180/n)))

Simplified:

a = 500 * ((500/n) * tan(90-(180/n)))

Multiplied out brackets:

a = 250000/n * tan(90-(180/n)))

I used the formulas above to see what happens as you increase the number of sides.  The results are as followed:

Number of sides

area/m²

10

76942.09

20

78921.89

30

79286.37

40

79413.78

50

79472.72

60

79504.74

70

79524.04

80

79536.56

90

79545.15

100

79551.29

110

79555.83

120

79559.29

130

79561.98

140

79564.11

150

79565.84

160

79567.24

170

79568.41

180

79569.39

190

79570.22

200

79570.93

210

79571.53

220

79572.06

230

79572.52

240

79572.93

250

79573.28

260

79573.60

270

79573.88

280

79574.13

290

79574.36

300

79574.56

From this results table, I can produce a graph of area against number of sides.  From the graph and table, I can see that as you increase the number of sides, the area increase.  Therefore, the shape that has the maximum area, with a perimeter of 1000m, is a circle because it has infinite sides.  As n increases and approaches infinite, the shape should become a circle.  Now I am going to investigate a circle and see if this is true.

Circle

a = area of circle

c = circumference of circle

r = radius of circle

a         =

c        = 2r

If the circumference was 1000m:

1000m        = 2r

1000/2        = r

Simplified:

500/        = r

Substitute r in the formula to work out the area of a circle:

a        = *(500/

a        = *(500/)*(500/)

a        = *(250000/²)

Simplified:

a        = 250000/

a        = 79577.47m²

The area of the circle is larger than the 30 side shape because a circle has infinite sides.

Conclusion

For any given n-sided shape, regular shapes have the maximum area and as the number of sides of a shape increase, the area increases.  A circle has the maximum area because it has infinite sides.  Therefore the farmer should fence off a circle with a perimeter 1000m because it would create an area of 79577.47m².

...read more.

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