The Fencing Problem - Math Coursework

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The Fencing Problem - Math

The task

--------

A farmer has exactly 1000m of fencing; with it she wishes to fence off

a level area of land. She is not concerned about the shape of the plot

but it must have perimeter of 1000m.

What she does wish to do is to fence off the plot of land which

contains the maximun area.

Investigate the shape/s of the plot of land that have the maximum

area.

Solution

--------

Firstly I will look at 3 common shapes. These will be:

------------------------------------------------------

[IMAGE]

A regular triangle for this task will have the following area:

1/2 b x h

1000m / 3 - 333.33

333.33 / 2 = 166.66

333.33² - 166.66² = 83331.11

Square root of 83331.11 = 288.67

288.67 x 166.66 = 48112.52²

[IMAGE]A regular square for this task will have the following area:

Each side = 250m

250m x 250m = 62500m²

[IMAGE] A regular circle with a circumference of 1000m would give an

area of:

Pi x 2 x r = circumference

Pi x 2 = circumference / r

Circumference / (Pi x 2) = r

Area = Pi x r²

Area = Pi x (Circumference / (Pi x 2)) ²

Pi x (1000m / (pi x 2)) ² = 79577.45m²

I predict that for regular shapes the more sides the shape has the

higher the area is. A circle has infinite sides in theory so I will

expect this to be of the highest area.

The above only tells us about regular shapes I still haven't worked

out what the ideal shape is.

Width (m)

Length (m)

Perimeter (m)

Area (m²)

500

0

1000

0

490

10

1000

4900

480

20

1000

9600

470

30

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