Fencing problem

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Maths Coursework                                           SSMC                               Nicholas Thorburn

Fencing problem

I realise intuitively that a circle will cover the largest area, but it is hard to prove it. I decided to do this logically, according to the sheet, by looking at the area that can be covered by a rectangle. I decided to produce a table, and then a graph, showing varying lengths and breadths in a logical manner, decreasing one by 50, and increasing the other by 50.

These are my preliminary results:

I produced a graph showing area from this:

You can see that this graph is a parabola, so the highest point will indicate the greatest area, which is 62,500. I have put the base along the x axis because you can work out the other two sides of a rectangle from one side (multiply X2, then you will be left with a number, divide it by two to get the other side). This is so that I can show what the corresponding measurements are for each point on my graph. When I narrow my search down to rises and falls of 12.5, I obtained these results:

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This shows that the greatest area, which can be covered by a RECTANGLE, is 62343.75sq2m. A width of 262.5 and a length of 237.5 obtain this. The greatest area covered by a four-sided shape however, is a square, 250X250, which covers 62,500 sq2m.

Triangles

        I am only going to use isosceles triangles. This is because if know the base length, then I can work out the other 2 lengths, because they are the same. If the base is 200m long then I can subtract that from 1000 and divide it by 2. This means that I can say that ...

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