Math Coursework Fencing

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The Fencing Problem

Introduction

The fencing problem involves finding a solution for a farmer who wants to fence a levelled piece of land. The farmer has exactly 1000m of fencing therefore he wants to achieve the maximum possible area. This will require me to test all shapes and see which shape gives the biggest area. The main shapes that I will look at are the following;

  1. Rectangles
  2. Trapeziums
  3. Parallelograms
  4. Kites
  5. Triangles
  6. Polygons greater than 4 sides
  7. Circles

Rectangles

In , a rectangle is defined as a  where all four of its angles are .

From this definition, it follows that a rectangle has two pairs of parallel sides of equal ; that is, a rectangle is a . A  is a special kind of rectangle where all four sides have equal length; that is, a square is both a rectangle and a .

The  of a rectangle is the  of its length and its width; in symbols,. For example, the area of a rectangle with a length of 5 and a width of 4 would be 20, because 5 × 4 = 20.
I first started by drawing two rectangles and finding out their area using the formula:

A table will be best to record my results

The table shows that the rectangle with the maximum area, 62500m2  has lengths of 250 and widths of 250. Therefore this regular rectangle is a square. The values in this table can be used to create a quadratic graph which proves that the maximum area occurs when the length is regular.

This quadratic graph  has a maximum which occurs when the length is 250 metres, therefore this has to be the length which produces the maximum area.

Trapeziums

 A trapezium is a  two of whose sides are  to each other. Some authors define it as a quadrilateral having exactly one pair of parallel sides, so as to exclude .

In an , the base angles are congruent, and so are the pair of non-parallel opposite sides.

Thus, if a and b are the two parallel sides and h is the distance (height) between the parallels, the area formula is as follows:

I started by drawing an isosceles trapezium and finding out its area

                                a

        c            d

                                

Because it is an isosceles trapezium c and d will be equal.        

        50m

       

        a

        250m        250m

        c        

        450m

To find height

To find 2x

If a < b, do b-a

If a > b, do a-b        

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2x=b-a

2x=450-50        x        x


h        250

200

Apply Pythagoras theorem to find height









Apply trapezium formulae to find area:




It is clear that a table I necessary as it will allow me to work out the best trapezium in the quickest manner:

The table shows that the trapezium with the maximum area, 62500m2 is achieved when a=250 and b=250. Therefore this isosceles trapezium is a square.

Shown geometrically:

        Transition

        Giving bigger area

Parallelograms

A parallelogram is ...

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