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  • Level: GCSE
  • Subject: Maths
  • Word count: 1631

The Fencing Problem

Extracts from this document...

Introduction

7/7/02The Fencing Problem        Clare Dutton

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

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

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

                   Investigate the shape, or shapes, that could be used to fence in the maximum area using exactly 1000m of fencing each time.

Background work

Before I look for the answer, I will do some trial investigations to determine a good way of investigating which shape gives the maximum area.

Do regular or irregular shapes give a larger area?

Triangle

I know that the maximum area for an equilateral triangle (regular) is

48112.5224 m² (4.dp), by using ½ absinc.

                                  1 side: 1000÷3=333⅓

                                              Area (A) = ½ absinc

                                    A= (333⅓ × 333⅓ × sin60) ÷2

A=48112.522432468813709095731708496

                            A=48112.5224 m² (4.dp)

An example of an irregular triangle is an isosceles triangle. To find the area of one it must be divided in two and Pythagoras’s theorem and ½ b×h must be used.

e.g.

             Base=50m

Other 2 sides= (1000-50) ÷2

                   =475m

 b²=h²-a²

 b²=475²-25²

b²=225625-625

b²=225000

 b=√225000

b=474.341649m

 Area=½b×h

A= (474.341649×50) ÷2

       A=11858.54123m²

Here are some more examples:

Base

Other 2 Sides

Height (b²=h²-a²)

Area=½b×h

100

450

447.21359549995800

22360.679774997900

150

425

418.33001326703800

31374.750995027800

200

400

387.29833462074200

38729.833462074100

250

375

353.55339059327400

44194.173824159200

300

350

316.22776601683800

47434.164902525700

350

325

273.86127875258300

47925.723781702000

400

300

223.60679774997900

44721.359549995800

450

275

158.11388300841900

35575.623676894300

499

251

22.36067977499790

5578.989603861980

500

250

0

0

When the size of the base reaches 500m it is no longer possible to create an isosceles triangle because they would be inverted. The largest isosceles triangle is one with a base of 332m. This is because it is closest to the side length of an equilateral triangle.  This is shown here:

Base

Other 2 Sides

Height (b²=h²-a²)

Area=½b×h

320

340

300.00000000000000

48000.000000000000

330

335

291.54759474226500

48105.353132473700

332

334

289.82753492378900

48111.370797348900

333

334

288.96366553599800

48112.450311743600

333.3333333

333

288.67513459510200

48112.522432468800

...read more.

Middle

This shows that a regular shape has a larger area than an irregular shape.

Quadrilateral

 I know that a square (regular quadrilateral) has a maximum area of 62500m².

image00.png

                    1 side: 1000÷4=250m

                           250×250=62500

A=62500 m²

An example of an irregular quadrilateral is a rectangle.

Their areas can be found by using the formula: xy

e.g.

y=5m

x=(1000-2y)/2                        

   x=495m

A=x × y                

A= 495×5

A=2475 m²

image01.png

Here are some other examples:

y  in m

x in m

Area in

10

490

4900

20

480

9600

40

460

18400

60

440

26400

80

420

33600

100

400

40000

120

380

45600

140

360

50400

160

340

54400

180

320

57600

220

280

61600

260

240

62400

300

200

60000

380

120

45600

460

40

18400

498

2

996

As you can see in this table the rectangle with the largest area is the one with 260m and 240m as side lengths. These measurements are very close to those of the square. Even closer to the square’s measurements are 249m and 251m giving a maximum area of 62499m².

The square gives the larger area and so we can see that a regular shape is better than an irregular shape. Another quadrilateral is a trapezium but this does not have a shape similar to the square or rectangle and so cannot have a larger area.

image02.png

The trapezium (red) appears to have the same area as the square (black) if you take the overlap on the right and add it inside the square on the left. The perimeters are not the same though. Side x is longer than y and so the square needs a larger perimeter to make it 1000m. Therefore the square has a larger area than a trapezium. This will be the same with a rectangle and trapezium and a parallelogram.

Here is the rectangle fitting into the square:

image03.png

The rectangle overlaps the square on one side but leaves a gap on the other where the square is larger. You could argue that if you took the overlapping strip and placed it in the gap the areas would be the same. But there would still be a space (shown in blue) where the rectangle was smaller.

This proves that irregular shapes do not have a larger area than regular ones.

...read more.

Conclusion

                 A=48112.5224 m² (4.dp)

4 sides

b=1000÷4=250 m

÷2=125 m

360÷4=90°

÷2=45°

        Height of Triangle=125÷Tan45

    h=125 m

      Area of 1 triangle =½ b×h

      A=½ × 250 × 125

              A=15625 m²

(×4)       Area of Square= 62500m²

5 sides: Pentagon

   1 side: 1000 ÷5=200 m

              ÷2=100 m

   Angle at centre point: 360 ÷5=72°

                               ÷2=36°

Height=100/tan 36

           Height=1376.38192

       Area of triangle=½ b×h

    Area of 1 triangle=137638.192m²

    (×5)        Area of Pentagon=688190.96m²

6 sides: Hexagon1 side: 1000÷6=166.6666667 m

          ÷2=83.33333335 m

                 1 angle: 360÷6= 60°

                                     ÷2= 30°

 Height=83.33333335/ tan30

                           Height= 144.3375673 m

     Area of triangle = ½ b×h

A=83.33333335×144.3375673 m²

(×6)       Area of hexagon=12028.13061 m²

7 Sides: Heptagon

1000÷7=142.8571429 m

÷2=71.42857145 m

360÷7=51.42857143°

÷2=25.71428572°

h=71.42857145÷Tan 25.71428572

h=148.3229569 m

   Area of triangle=½ b×h

A=10594.49692 m²

(×7)        Area of Heptagon= 74161.478m²

8 Sides: Octagon

  1000÷8=125 m

  ÷2=62.5 m

    360÷8=45°

÷2=22.5°

         h=62.5÷Tan 22.5

h=150.8883476 m

Area of triangle=h ×62.5

A=9430.5217525 m²

(×8)       Area of Octagon= 75444.1738 m²

9 Sides: Nonagon

1000 ÷9=111.1111111 m

÷2=55.55555555 m

360 ÷9= 40°

÷2= 20°

h=55.55555555 ÷Tan 20

=152.6376344m

Area of triangle=152.6376344×55.55555555

                                                       A=8479.86858 m²

(×9)      Area of Nonagon= 76318.81722 m²

10 sides: Decagon

1000÷10=100 m

          ÷2=50 m

360÷10=36°

÷2=18°

 h=50÷Tan 18

h=153.8841769 m

Area of triangle= 153.8841769 × 50

      A=7964.208845m²

(×10)       Area of Decagon= 76942.08845 m²

11 sides: Undecagon

1000÷11=90.90909091 m

          ÷2=45.45454546 m

360÷11=32.72727273°

        ÷2=16.36363637°

h=45.45454546÷Tan 16.36363637

h=154.8039654 m

Area of triangle=½ b×h

                      A=7036.54388m²

(×11) Area of Undecagon= 77401.98268 m²

12 sides: Dodecagon

1000 ÷12=83.33333333 m

           ÷2=41.66666667 m

360 ÷12=30°

         ÷2=15°

h=41.66666667÷Tan 15

h=155.502117 m

Area of triangle=½ b×h

                      A=6479.254876 m²

(×12)       Area of Dodecagon= 7751.05851 m²

13 sides: Tridecagon

1000÷13=76.92302692 m

          ÷2= 38.46153846 m

360÷13= 27.71428571°

        ÷2=13.84615385°

h=38.46153846÷Tan13.84615385

h=155.8620234 m

Area of triangle=½ b×h

                      A=5994.693207 m²

(×13) Area of Dodecagon=77931.01169m²

14 sides: Tetra-decagon

1000÷14=71.42857143 m

          ÷2=35.71428572 m

360÷14= 25.71428571°

        ÷2=12.85714286°

h=35.71428572÷Tan 12.85714286

h=156.4745095 m

Area of triangle=½ b×h

                      A=156.4745095×35.71428572 m²

(×14) Area of Tetra-decagon= 78237.25476 m²

...read more.

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