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The Fencing Problem.

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Introduction

Yonathan Maalo

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.

Aim

The aim of the investigation is to find out which shape or shapes of fencing will cover the maximum area of land using exactly 1000 metres of fencing material.  

Prediction

I am predicting

...read more.

Middle

58,094.75019

8

Parallelogram

51,961.5

11

Parallelogram

34,641.01615

21

Parallelogram

45,466.3337

16

Rhombus

54,126.59

10

Rectangle

40,000

18

Rectangle

22,500

23

Circle

79,577.47

1

Kite

56,291.643

9

Kite

32,475.95264

23

Kite

47,631.39721

13

Right-angled Triangle

37,500

20

Isosceles Triangle

44,721.36

17

...read more.

Conclusion

All the four-sided shapes (parallelogram, rhombus, rectangle, square and kite) had covered similar areas. In general, the four-sided shapes covered areas between 51,000 and 63,000 square metres (Table 1 and Figs 1 & 2). The square shaped fence covered the greatest area, 62,500 square metres, compared with the other four-sided shapes.

Conclusion

The maximum area of land covered with the 1000m perimeter fencing was achieved by using the circular fencing. The maximum area covered was 79,577.47m2. The area of land covered appeared to increase with increase in the number of sides of the given shape of fencing material as well as shapes that appeared wider.

...read more.

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