The fencing problem.

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A farmer has exactly 1000 metres of fencing, with it she wishes to fence off a plot of level land. She is not concerned about the shape of the plot, but it must have a perimeter of 1000 metres. I am going to consider using rectangles first as this shape is very simple and it would be easy for me to work out the maximum area and perimeter.

Rectangle 5 is a square. After this the order of the rectangles are the same as before but in a reverse order, this is shown in rectangle 6. To work out the area of each rectangle I am going to multiply the length of the rectangle by the width of the rectangle. Here is a table to show you my results.

Length (m)

Width (m)

Area (m )

450

50

22500

400

00

40000

350

50

52500

300

200

60000

250

250

62500

200

300

60000

50

350

52000

00

400

40000

50

450

22500

The square gives me the maximum area I have highlighted this in my results table in red. I have produced a graph to show you my results.

I am now going to consider using triangles. I am going to find out the maximum area and the most suitable triangle to do this. To find out the most suitable triangle I am going to use a string. To do this I am going to tie the ends together, from this I am going to keep the base the same and ask my partner to move it into a triangle and ask her to change it until I see a triangle, which is suitable.

Picture 1

In picture one I have shown you how I have found the isosceles triangles as I think this kind of triangle will give me the maximum area whereas a scalene triangle would not.

475m 475m 450m 450m 425m 425m

50m

100m 150m

400m 400m

375m 375m

350m 350m

200m

250m

300m

333.3m 333.3m

325m 325m

333.3m

350m

In triangle seven it shows us an equilateral triangle. This is where I should stop as I am just investigating isosceles triangles. To find out the area of each triangle I am going to half the base and multiply it with the height. As I am not given the height I am going to have to fine the height by using Pythagoras theorem. Here are three examples.
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475m 475m ? 475m

50m 25m

425m 425m ? 425m

?

150m 150m

375m 375m ? 375m

Here is a table to show you my results.

Base (m)

Slant Height (m)

Height (m)

Area (m )

50

475

474.342

1858.55

00

450

474.214

22360.7

50

425

418.330

31374.75

200

400

387.298

38729.8

250

375

353.553

44194.125

300

350

316.228

47434.2

333.3
...

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