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• Level: GCSE
• Subject: Maths
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# the fencing problem

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Introduction

The Fencing Problem

A farmer wants to fence a plot of level land and has exactly 1000 meters of fencing. The Farmer wants to have a perimeter of 1000m but is not concerned about the shape of the plot. I am going to investigate the maximum area of a land using different families of shapes. I will investigate all families of shapes and see which one has the maximum area. I will be using rectangles, triangles and polygons.

First I will be investigating Quadrilaterals and I will be using rectangles to find the maximum area using different lengths and keeping the perimeter the same.

Perimeter= Width+Length+Width+Length

= 250+250+250+250

= 1000m

Area of Rectangle= Width*Length

= 250*250

= 62500m²

 Width (m) Lengths (m) Perimeter (m) Area (m²) 450 50 1000 22500 400 100 1000 40000 350 150 1000 52500 300 200 1000 60000 250 250 1000 62500 200 300 1000 60000 150 350 1000 52000 100 400 1000 40000 50 450 1000 22500

In my investigation I have found that the maximum area is 62500m² with the lengths 250 and 250. In my search I went down by 50 for the Width and up by 50 for the length so that the width and length add up to 1000 meters. think that this is not exactly the maximum area so I will refine my search by doing a decimal search.

 Width (m) Lengths (m) Perimeter (m) Area (m²) 249.5 250.5 1000 62499.75 249.6 250.4 1000 62499.84 249.7 250.3 1000 62499.91 249.8 250.2 1000 62499.96 249.9 250.1 1000 62499.99 250 250 1000 625000

After refining my search into a decimal search this table can tell me that the maximum area stayed the same at 62500m². I went up by 0.

Middle

302.5

395

229.1

45252

1000

300

300

400

223.6

44721

1000

In this Search I went down by 2.5 for the lengths and up by 5 for the base

so that both lengths and base add up to 1000 meters. From this table I

can tell that in my first decimal search the maximum is 48110m², which I don’t think is as high as I could get it, therefore I will refine my search.

 Length2(m) Length2(m) Base(m) Height(m) Area(m²) Perimeter(m) 335 335 330 291.5 48105 1000 334.5 334.5 331 290.6 48109 1000 334 334 332 289.8 48111 1000 333.5 333.5 333 288.9 48112 1000 333 333 334 288.0 48112 1000 332.5 332.5 335 287.2 48110 1000 332 332 336 286.3 48107 1000 331.5 331.5 337 285.4 48103 1000 331 331 338 284.6 48098 1000 330.5 330.5 339 283.7 48091 1000 330 330 340 282.8 48083 1000

In this Search I went down by 0.5 for the lengths and up by 1 for the base so that both the lengths and base add up to 1000 meters. From this table I can tell that there are two triangles with the same maximum area of 48112m². I will refine my search even further to find the exact maximum area for the triangles.

 Length2(m) Length2(m) Base(m) Height(m) Area(m²) Perimeter(m) 333 333 334 288.0 48096 1000 333.1 333.1 333.8 288.2 48100 1000 333.2 333.2 333.6 288.4 48105 1000 333.3 333.3 333.4 288.6 48109 1000 333.4 333.4 333.2 288.7 48097 1000 333.5 333.5 333 288.9 48101 1000

In this Search I went up by  0.1 for the lengths and down 0.2 for the base so that both the lengths and base add up to 1000 meters.

From this table I can say that the maximum area is 48109m² , with the lengths 333.3,333.3 and base 333.4. I will refine my search even further with two numbers behind the decimal.

 Length2(m) Length2(m) Base(m) Height(m) Area(m²) Perimeter(m) 333.31 333.31 333.38 288.63 48111.7347 1000 333.32 333.32 333.36 288.65 48112.182 1000 333.33 333.33 333.34 288.67 48112.6289 1000 333.34 333.34 333.32 288.66 48108.0756 1000

In this Search I went up by 0.01 for the lengths  and down 0.02 for the base so that both the lengths and base add up to 1000 meters.

After all my investigation of triangles I have found that the highest maximum area for the triangles is 48112m², with the lengths 333.33, 333.33 and base 333.34. The shape of this triangle is isosceles.

Next i will be investigating polygons, I will be investigating the pentagon, hexagon, heptagon, and octagon. I want to investigate these shapes because I think that after going through the rectangles and triangles and investigating them and their sides I have decided to do theses shapes as all of them have different number of sides.

Pentagon:

200m

360°/5 = 72°

1 Triangle of Pentagon = 72°/ 2= 36°

Base = 1000/5

= 200m

Half the base = 100m

Height of triangle=

Tan36°= 100/height  :cross multiply

= Height =100/Tan 36°

= 137.64m

Area of triangle = (Base* Height) /2

= (200*137.64)/2

= 13764m²

Area of Pentagon =  Area of 1 triangle*5

= 13764*5

= 68820m²

In my investigation I have found that in the family of polygons the pentagon can give the maximum area of 68820m².

To investigate the n sides I have to first find a suitable formula which can help me find the area of a lot of n sides.

Tan (360/(2*n) = Tan(180/n)

Base = 1000/n

½Base = 1000/(n*2)

Tan(180/n) = (1000/2n)/h

H = (1000/2n)/Tan(180/n)

A = (1000/n)*((1000/2n)/Tan(180/n))/2

= (250000/n)*1/Tan (180/n)

 Number of sides Area 5 68819.09602 6 72168.78365 7 74161.47845 8 75444.17382 9 76318.81721

Conclusion

Circle:

Area of Circle = Πr²

Π = 3.14

= 159.24m

Area of Circle = Πr²

= Π*(159.24)²

= 79622.16566m²

In my investigation I have found that the maximum area for the circles is 79622.16566m². I also found that the highest maximum area was for the circle after all the different shapes I investigated.

I have also noticed that the limit tot he area of the polygons is just a bit smaller than the maximum area of the circle.

 Shape Sides Area (m²) Triangle 3 62500 Quadrilaterals 4 48112 Pentagon 5 68820 Hexagon 6 72192 Heptagon 7 74249 Octagon 8 75448 Circle 79622

This table tells me that the highest maximum area is for the circle. After doing my investigation I went over the maximum areas for each family of shapes and compared them. For the Rectangles the maximum area I got was 62500m² which was a square. For the triangles the maximum area I got  was 48112m²I, which was a isosceles triangle.

For the polygons I used four different shapes which were a pentagon,  a hexagon, a heptagon and a octagon For the Pentagon the maximum area I got was 68820m². For the Hexagon the maximum area I got was 72192m². For the Heptagon the maximum area I got was 74249m². And for the octagon the maximum area I got was 75448m². I also did circles and found that the maximum area was 79622.16566m², which was the highest maximum area overall. I think the farmer should use the shape circle to fence his 1000 meter land.

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

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