The Fencing Problem - Mathematics

The Fencing Problem

Introduction

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I have been given 1000 meters of fencing and my aim is to find out the

maximum area inside.

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Prediction

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I would predict that the more sides the shape has, then possibly the

bigger the area it will have, although I have nothing to base this on,

it will be what I am about to investigate.

Shapes:

I am going to start with the rectangle, I think this is a good

starting block because I am able to vary the widths and lengths to see

which has the bigger area. If I discover that the rectangles with

equal sides i.e. square bring me the best result, then I will try to

direct my investigation into furthering that particular theory.

Rectangles

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[IMAGE]

Area = 40 000 m2

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[IMAGE]

Area = 60 000 m2

[IMAGE]

Area = 62 500 m2

It appears that the square shape has a bigger area, I would possibly

say that this is because the square has two bigger numbers, which are

multiplied together to give a greater number than when a big number is

multiplied with a smaller number.

However, I cannot take this for granted and I think using one more

shape will be useful in order to back up my theory.

[IMAGE]

Area = 52 500m

This proves my theory regarding squares and I shall now put my results

into a graph to show what I have found.

Length (m) Width (m) Area (m)

400 100 40 000

300 200 60 000

250 250 62 500

150 350 52 500

I will now further my investigation by looking at shapes of a

different nature:

[IMAGE]

Regular Pentagon

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The regular pentagon has 5 sides, and as we get 1000m of fencing, this

means each side will be 200m (1000¸5=200).