Mathematics

Looking at Different Shapes - Math Problem

For this investigation, I will be looking at different shapes, and the

areas the different shapes give. The exact question is:

'A farmer has exactly 1000m of fencing, and wants 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 1000m. She wishes to fence of the area of

land, which contains the maximum area.

Investigate the shapes that could be used to fence in the maximum area

using exactly 1000m of fencing each time.

I will show the working in the form of formulas, putting results in

tables, and then transfer the tables into graphs. Once this is

completed, I will draw up a conclusion.

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Prediction

My prediction is that as the number of sides increase, as will the

area. I think this as the area of a rectangle, or any other

quadrilateral, will have a bigger area than a triangle when using the

same perimeter. I have no reason not to believe that this pattern of

increasing sides/increasing area will continue.

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I am going to start investigating different shape rectangles, all

which have a perimeter of 1000m. Below are 4 rectangles (not to scale)

showing how different shapes with the same perimeter can have

different areas.

In a rectangle, any 2 different length sides will add up to 500,

because each side has an opposite with the same length. Therefore in a

rectangle of 100m X 400m, there are two sides opposite each other that

are 100m long and 2 sides next to them that are opposite each other

that are 400m long. This means that you can work out the area if you

only have the length of one side. To work out the area of a rectangle

with a base length of 200m, I subtract 200 from 500, giving 300 and

then times 200 by 300. I can put this into an equation form.

1000 = x (500 - x)

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300

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Using this formula I can draw a graph of base length against area. (on

3rd page)

According to the table and the graph, the rectangle with a base of

250m has the greatest area. This shape is also called a square, or a

regular quadrilateral. Because I only measured to the nearest 10m, I

cannot tell whether the graph is true, and does not go up just to the

sides of 250m. I will work out the results using 249m, 249.5 and

249.75

Base (m)

Height (m)

Area (m2)

249

251

62499

249.5

250.5

62499.75

24975

250.25

6249993.75

250

250

62500

250.25

249.75

62499.9375

250.5

249.5

62499.75

251

249

62499

All of these results fit into the graph line that I have, making my

graph reliable.

Now that I have found that a square has the greatest area of the

rectangles group, I am going to find the triangle with the largest

area. Because in any scalene triangle, there is more than 1 variable,

there are countless combinations, so 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:

[IMAGE][IMAGE]Side = (1000 - 200) Ð"Ñ"Ð'* 2 = 400

400

400

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h

100

100

To work