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• Level: GCSE
• Subject: Maths
• Word count: 2944

# 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.

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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