The Fencing Problem

These are the results from my investigation

a b Area = 500(500-a) (500-b) (500-c)

c

Area = 500(500-333 1/3) (500-333 1/3) (500-333 1/3)

=83334.248

Area = 500(500-300) (500-350) (500-350)

=47434.164

Area = 500(500-250) (500-450) (500-300)

=35355.339

Area = 500(500-250) (500-500) (500-250)

=6123.724

Area = 500(500-150) (500-500) (500-300)

=5916.079

Area = 500(500-400) (500-400) (500-200)

=38729.833

Area = 500(500-450) (500-450) (500-150)

=20916.501

Now I will try a different family of shapes a 4 sided shape but I think that the greatest area of a shape will come from a regular shape.

Area = 250 x 250

= 62,500m2

Area = 200 x 300

= 60,000m2

Area = 150 x 350

= 52,500m2

Area = 100 x 400

= 40,000m2

Area = 50 x 450

= 22,500m2

From this I can see that there is a pattern and this is that the regular shapes have the highest for that specific area. I will now increase the number of sides on a shape and see if I can find a maximum area.

I can see that as the number of sides increase the area also increases. The question now is does a circle have infinity sides, 1 single side or a peak number of sides

n- sided

Area =

This formula shows that as n increases to infinity approaches tan 90 so that the angle between adjacent sides becomes 2x 90= 180o thus giving a circle.

For a circle of a radius r =