I will double check again to be certain that I still get maximum at 250m length by 250m.I will use a pitch of 0.1m between 249.0 and 250.2.
Using Microsoft Excel, I created a graph from my results, this graph shows me that the as the length increases so does the area, but this is only true until a maximum area is reached. After this maximum point has been reached the length continues to increase while the area begins to decrease. This final graph tells me that 250m is the maximum peak of a rectangle.
Conclusion:
These tables and graphs all showed me that the maximum area was 62500m and the highest peak is at 250m. So if a 1000m fence is used these dimensions are the maximum for a rectangle which is really a square. So when the shape has equal sides it has maximum area. I will look at some other quadrilaterals such as rhombuses, parallelograms and trapeziums.
This is a parallelogram:
This becomes a rectangle but my previous work has shown that a rectangle doesn’t give me the maximum area.
This is a Trapezium:
This also becomes a rectangle so I know that its area will be less than 62500m
This is a rhombus:
This rhombus will give the maximum area of 62500m because it can be transformed into a square which has equal sides.
Conclusion:
So far only a quadrilateral with equal sides has given me maximum area. In the next section I will look at different kinds of triangles. I will look at isosceles, equilateral, scalene and right angled triangles.
Triangles
Now that I have investigated the quadrilaterals I will start with triangles. I will try to do different sorts of triangles. I will investigate the highest area using only 1000m of fencing I will start with an isosceles triangle. An isosceles triangle has two sides which are the same length, and a third side which is not the same length. I am going to use the Pythagoras theorem on the isosceles triangle. As the Pythagoras rule can only be used in right angled triangles (which an isosceles is not) I will have to half the triangle.
I will start with a base of 50m.
1000-50=950 = 475m
2
475m
50m h 475
1000-100=900=450m
2
25m
450m
h
100 m
50m
I will out my calculations in a table using excel spreadsheet.
This triangle would not be possible because the sides would not be able to join up.
This table shows that 300 is the maximum an isosceles triangle will go up to because 350 will not make a triangle. So I will try to find out if 300 base is the real maximum. I will do this by uinvestigating around the numbers 300 to 349. but firstly I will put all rhis in a graph using excel spreadsheet.
Now I will draw more tables and graphs finding a maximum area.
I will draw more table and graphs investigating around 300 to 350.
The maximum area has gone up to 335 but I will investigate further by
going up in ones between 330 and 340. I will use a excel to draw another
table. And see if I can find a more accurate result b3tween those two
numbers.
In this table I have found a more accurate result but I am still not satisfied that this is the maximum so I will go on to investigate between 333.1 to 334. which means that I will draw another table using excel spreadsheet again. This will be my final table on isosceles triangles because after this I will know the maximum area for an isosceles triangle.
This final table shows me that 333.3 is the maximum and 48112.52 is the maximum area. it seems the area keeps increasing and my isosceles triangle is beginning to look like an equilateral triangle. The red coloured boxes are the triangles which would not be possible because the base is too long so the sides’ wont join together to make a triangle as they would not be able to reach each other.
Observation:
1. I noticed that as I increased the base the other 2 sides became shorter because
My fencing is a fixed amount of 1000m.when the sides are 330 the other side Becomes longer in length.
2. Now I will move on to scalene triangles to see if they will give me a higher area. I will need to use hero’s formula.
