# A farmer has exactly 1000m of fencing and wants to fence off a plot of level land. She is not concerned about the shape of the plot. She wishes to fence off the plot of land which contains the maximum area. I will be investigating the shape

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Introduction

A farmer has exactly 1000m of fencing and wants to fence off a plot of level land. She is not concerned about the shape of the plot. She wishes to fence off the plot of land which contains the maximum area. I will be investigating the shape, or shapes that could be used to fence in the maximum area using exactly 1000m of fencing each time.

I will attempt to work this problem out using a logical sequence of thought. So I will first investigate the shape that is simplest to work out the area of, rectangles.

The formula to work out rectangles is width*height

I will now show a number of rectangles to show how to work out the areas. I will fill the values for height width and area by hand.

Here is the table of rectangle area’s I made using excel. I’ve cut it down to show the first part of the table up to an area of 16,000 cm2.

width (A) | Height (B) | Area [C] |

10 | 990 | 9900 |

20 | 980 | 19600 |

30 | 970 | 29100 |

40 | 960 | 38400 |

50 | 950 | 47500 |

60 | 940 | 56400 |

70 | 930 | 65100 |

80 | 920 | 73600 |

90 | 910 | 81900 |

100 | 900 | 90000 |

110 | 890 | 97900 |

120 | 880 | 105600 |

130 | 870 | 113100 |

140 | 860 | 120400 |

150 | 850 | 127500 |

160 | 840 | 134400 |

170 | 830 | 141100 |

180 | 820 | 147600 |

190 | 810 | 153900 |

200 | 800 | 160000 |

From my set of result’s I’ve made a table which shows a interesting result, has a perfect line of symmetry down the middle and the results which we began with we also end with.

Middle

The graph of the scalene triangles, has a line of symmetry shown

As you can see the maximum value highlighted below, show’s the shape of an isosceles triangle. Therefore I will be doing isosceles triangles next.

I’ll start my investigation of isosceles triangles by working out the area of a few triangles. Then I will show my results in the form of a table and analyse them.

Answer = 11858.54cm2

Answer = 22360.68cm2

Answer = 31374.75cm2

Answer = 38729.83cm2

Answer =44194.17cm2

Answer =47434.16cm2

Answer =47925.72cm2

Answer =44721.36cm2

Answer =35575.62cm2

I will be using hero’s formula to find the area of isosceles triangles. Here is my table.

BASE [C] | SIDE 1 (A) | SIDE 2 (B) | Perimeter | S | S-A | S-B | S-C | AREA |

200 | 400 | 400 | 1000 | 500 | 100 | 100 | 300 | 38729.83346 |

210 | 395 | 395 | 1000 | 500 | 105 | 105 | 290 | 39982.80881 |

220 | 390 | 390 | 1000 | 500 | 110 | 110 | 280 | 41158.23125 |

230 | 385 | 385 | 1000 | 500 | 115 | 115 | 270 | 42253.69806 |

240 | 380 | 380 | 1000 | 500 | 120 | 120 | 260 | 43266.61531 |

250 | 375 | 375 | 1000 | 500 | 125 | 125 | 250 | 44194.17382 |

260 | 370 | 370 | 1000 | 500 | 130 | 130 | 240 | 45033.321 |

270 | 365 | 365 | 1000 | 500 | 135 | 135 | 230 | 45780.72739 |

280 | 360 | 360 | 1000 | 500 | 140 | 140 | 220 | 46432.74706 |

290 | 355 | 355 | 1000 | 500 | 145 | 145 | 210 | 46985.37006 |

300 | 350 | 350 | 1000 | 500 | 150 | 150 | 200 | 47434.1649 |

310 | 345 | 345 | 1000 | 500 | 155 | 155 | 190 | 47774.20852 |

320 | 340 | 340 | 1000 | 500 | 160 | 160 | 180 | 48000 |

330 | 335 | 335 | 1000 | 500 | 165 | 165 | 170 | 48105.35313 |

340 | 330 | 330 | 1000 | 500 | 170 | 170 | 160 | 48083.26112 |

350 | 325 | 325 | 1000 | 500 | 175 | 175 | 150 | 47925.72378 |

360 | 320 | 320 | 1000 | 500 | 180 | 180 | 140 | 47623.5236 |

370 | 315 | 315 | 1000 | 500 | 185 | 185 | 130 | 47165.9305 |

380 | 310 | 310 | 1000 | 500 | 190 | 190 | 120 | 46540.30511 |

390 | 305 | 305 | 1000 | 500 | 195 | 195 | 110 | 45731.55366 |

400 | 300 | 300 | 1000 | 500 | 200 | 200 | 100 | 44721.35955 |

410 | 295 | 295 | 1000 | 500 | 205 | 205 | 90 | 43487.06704 |

420 | 290 | 290 | 1000 | 500 | 210 | 210 | 80 | 42000 |

430 | 285 | 285 | 1000 | 500 | 215 | 215 | 70 | 40222.81691 |

440 | 280 | 280 | 1000 | 500 | 220 | 220 | 60 | 38105.11777 |

450 | 275 | 275 | 1000 | 500 | 225 | 225 | 50 | 35575.62368 |

460 | 270 | 270 | 1000 | 500 | 230 | 230 | 40 | 32526.91193 |

470 | 265 | 265 | 1000 | 500 | 235 | 235 | 30 | 28781.50448 |

480 | 260 | 260 | 1000 | 500 | 240 | 240 | 20 | 24000 |

490 | 255 | 255 | 1000 | 500 | 245 | 245 | 10 | 17324.11614 |

Conclusion

#

Now I can present the formula not in its finished state, but a usable one, the penultimate stage. I’ve cancelled out the like terms and I’m left with only a few more things to simplify.

I will now do the finished version of the formula and then test it out on 3 n-gon shapes.

Now to test the general formula, I will at first use the shapes which areas I’ve found out.

Octagons – 25000/8TAN(180/8)= 75444.174cm2

Nonagons – 25000/9TAN(180/9)= 76318.817cm2

Decagons (10 sided) – 250000/10TAN(180/10)= 76942.088cm2

50 sided shape - 250000/50TAN(180/50)= 79472.724cm2

Now here is a table containing the areas

Triangle | 48112.52243 | |

Square | 62500 | |

Pentagon | 68819.09602 | |

Hexagon | 72168.78365 | |

Heptagon | 74161.47845 | |

Octagon | 75444.17382 | |

nonagon | 76,318.817 | |

decagon | 76942.088 | |

dodecagon | 77751.05849 |

My graph show’s a general increase as the sides increase reaffirming my prediction that the area would increase as the sides went up. Although to show this theory working in a larger scope here is a table of areas increasing in 50’s.

50 sided shape | 79,472.72 |

100 sided shape | 79551.28988 |

150 sided shape | 79565.83568 |

200 sided shape | 79570.92645 |

250 sided shape | 79573.28271 |

The graph of the shapes going up in 50’s shows the same trend as before

After my research, I’ve concluded that the circle with a perimeter of 1000m is the largest area, and the farmer, if she wishes to have the largest area, should use this shape, although since circles do not tessellate, nor is it easy to make this shape on a large scale, she should use a simpler shape.

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

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