Fencing Problem

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John Smith        Thomas Alleynes High School        Maths Coursework

GCSE Maths Coursework-

The Fencing Problem

Introduction

I am going to investigate different a range of different sized shapes made out of exactly 1000 meters of fencing.  I am investigating these to see which one has the biggest area so a Farmer can fence her plot of land.  The farmer isn’t concerned about the shape of the plot, but it must have a perimeter of 1000 meters, however she wishes to fence off the plot of land in the shape with the maximum area.

Rectangles

I am going to look at different size rectangles to find which one has the biggest area.

Formula: Length x Width 

Table Of results

Conclusion

I have found that the four sided shape that had the biggest area when using 1000 meters of fencing, was a square with the measurement of 250m x 250m and the area=62500

Isosceles Triangles

I am now going to look at different size Isosceles triangles to find which one has the biggest area.  I am going to use Pythagoras Theorem to find the height of the triangle.

Pythagoras Theorem: a²=b²+c²

Formula To Find A Triangles Area: ½ x base x height

1. Base=100m        Sides=450m

 

Area: ½ x b x h

          ½ x 100 x 477

         =23850m²

2. Base=200m        Sides=400m

 

Area: ½ x b x h

          ½ x 200 x 387

         =38700m²

3. Base=300m        Sides=350m

Area: ½ x b x h

          ½ x 300 x 316

         =47400m²

4. Base=400m        Sides=300m

 

Area: ½ x b x h

          ½ x 400 x 224

         =44800m²

Conclusion

I have found that the three sided shape that had the biggest area when using 1000 meters of fencing, was a triangle with the measurements:

Length=300m  Sides=350m  Height=316m  Area=47400m²

Equilateral Triangle

I found that the biggest area of a triangle when the base goes up in hundreds each time was a triangle with the measurements:

Length=300m  Sides=350m  Height=316m.

I am now going to investigate an equilateral triangle to find out its area.

4. Base=333.3m        Sides=333.3m

 

Area: ½ x b x h

          ½ x 333.33 x 289

         =48166m²

Table Of results

Conclusion

I have found that the three sided shape that had the biggest area overall when using 1000 meters of fencing, was an equilateral triangle with the measurement:

Length=333.3m  Sides=333.3m  Height=289m and the area=48166

Pentagons

I am now going to look at a pentagon to find out its area when using 1000meters of fencing.

Formula To Find the Area Of A Pentagon:

Exterior Angle = 360°

                     No. Of Sides

Interior Angle = 180°-Exterior Angle

I will then use tangent of an angle, to work out the area of one of the five segments of the pentagon.  I will multiply this number by 5 because the 5 segments are all the same size.  Therefore I will get the overall area of the pentagon.

Pentagons Area

Area: Ex. Angle = 360° = 72°

                             5

             Int. Angle = 180°-72°= 108°

                       = 108°÷2=54°

Tan = opp

          adj

Join now!

Tan 54 = h

             100

1.376 x 100 = h

            138 = h

½ x 200 x 138 = 13800

13800 x 5 = 69000

Area = 69000m²

Hexagons

I am now going to look at a hexagon to find out its area when using 1000meters of fencing.

Formula To Find the Area Of A Hexagon:

Exterior Angle = 360°

                   No. Of Sides

Interior Angle = 180°-Exterior Angle

I will then use tangent of an angle, to work out the area of one of the six ...

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